OK... I did a lot of searching and found a few references on other MINI websites, but I never found a comprehensive guide on how to replace and bleed the clutch slave cylinder. This is a part that frequently fails on R53 MINIs (I'm on #4 in 4 years / 51k miles right now and carry a spare in the back of my car... I've sold this spare at cost twice to fellow motorists and had to use it once myself).
When you press the clutch petal, the master cylinder (attached to the clutch pedal arm) is compressed. This squeezes hydraulic fluid (in MINI's case, it's brake fluid) through a narrow pipe to the slave cylinder which then expands and moves a lever on the transmission housing to release the clutch.
How do I know that I have a bad slave cylinder?
The only sure way is to examine it and look for leaking fluid. The most regular place for this leak is under the expandable rubber boot, so you pry the wide end of it away and look under there. If you see fluid, you have a bad slave cylinder. I've also seen a leak at the fluid inlet side as well, which may require replacement of the tube upstream from the slave (under $20... a PITA to replace, but easily doable).
What would make me think that I've got a bad slave?
A bad slave is usually the culprit of a leaking clutch hydraulic system, but not always. A leaking clutch hydraulic system is usually found to give loose clutch feel in minor cases or a 'sinking clutch pedal' (the pedal doesn't return all the way up). You might have a hard time shifting the car into gear with the clutch pedal all the way down to the floor. One easy test is to stop the car on a level piece of pavement, push the clutch to the floor and put the car in 1st gear. Stay like this for 60 seconds. If you start rolling forward, you have a clutch fluid leak and I would strongly recommend that you examine the slave cylinder for leaking. You might also have a bad master cylinder (the one attached to the clutch pedal arm), but I haven't seen this problem nearly as often.
NOTE: If you experience these problems, get it checked out quickly if you don't feel confident to do it yourself. A sloppy feeling clutch pedal (hard to get the car into gear) can turn into an inoperable clutch pedal (CAN'T get the car into gear) in a short amount of time. You might be able to limp home or to your mechanic by leaving the car in neutral and pumping the clutch quickly 3-4 times before each shift, but this isn't going to last you long. A clutch hydraulic leak will only get worse, not better, and it will probably get worse very quickly (I got maybe 10-15 stop/start cycles with my last 2 instances using the pumping-the-pedal-then-shift trick and then the clutch was completely gone). Any cautious person would pull over and call for a tow. If you drive carefully, you may be able to limp the car to a mechanic. Think for yourself and do only what is safe.
The slave cylinder is located right behind the front bumper, on the driver's side of the car (between the driver's side fog light and the center license plate). It's a relatively inexpensive part ($38 or so with my local club discount - MINI Part 21-51-6-777-428 - Output CYM05C1). To do the replacement you will need a new part and the correct, fresh brake fluid for your car. I also like to use a pressure-bleeder (Motive Power Bleeder - European connection). You will need a jack and a jack stand, along with a handful of socket wrenches (a 10mm and 11mm deep socket for sure).
Slave Cyl. replacement and bleeding process:
NOTE: I am not a trained mechanic... I'm just handy around cars. This process has worked for me and I'm just trying to make someone else's life a bit easier. I'm always open to suggestions on a better process and corrections. Please think for yourself and always be safe!
1. Jack up the car under the driver's side front jack point and secure with a jack stand. I'll not rant and rave about the proper way to do this, but please be safe. It looks funny, but this corner is the only one you have to lift. It might give you more room to lift the other front corner of the car, but it wasn't necessary for me.
2. Remove the plastic shield that protects the underside of the engine. It should connect to the underside of the front bumper. I'd give you more help here, but I've replaced this part with an aluminum skid-plate, so I haven't seen the factory part in a while. I would guess that it will take 5-6 screws/bolts and some plastic clips to remove and it should be fairly easy to do.
3. Examine the old clutch slave. Most slaves fail by leaking fluid into the area protected by the plastic accordion boot. Pull the wide end of the boot away from the hard plastic slave cyl. and if brake fluid drips out, you have now confirmed that you have a problem here.
4. Prepare the new slave cyl. for bleeding. Get some duct tape (yes...tool of the gods) and tear off a nice 2-3 foot long strip. Place the tape sticky-side up on a table. Place the rounded end (compressible end) against the center of the tape and press down to squish the accordion completely. Lift the tape up and around the sides of the slave, wrapping it so that the piston stays completely compressed and you can access 1) the input hole and 2) the bleed valve easily.
Here's a pic of what the new slave cylinder should look like when taped up. Note that you can get to the bleed screw (far right), the fluid input connection (with white plastic protector in it) and that the piston is fully compressed (accordion dust boot on left is scrunched together):
5. Remove the 2 bolts that hold the old slave in place (10mm bolts). Tap it with the handle of your ratchet to slide it towards the front of the car and slip the slave cyl. out of it's holder.
6. Look at your new slave to figure out how the wire input retainer works (slides sideways). Pull this clip out on the old slave, pull it away from the input pipe and quickly place your finger over the hole (brake fluid is nasty stuff and oily, so don't wear your 'school clothing' or your mommy will be mad). The plastic connector on the input pipe has a small black washer that extends down to the tip of the input pipe (maybe 1/2 cm deep). If you don't see the washer there, look inside the old slave cyl. and dig around carefully... It's in there. Save the old washer and put it on the tip of the input pipe. Insert the new slave on to the input pipe and slide the locking clip over to stop the leaking fluid. Use a paper towel and a generic cleaner like 409 or Simple Green to clean up the mess (it will help you identify leaks later).
7. Fill your Motive power bleeder with fresh brake fluid (don't use old stuff) and apply 8-10 psi of pressure to the system. Attach a bleed bottle (or a hose leading into a catch pan) to the bleed valve on the new slave cyl. and open the valve. This is a 11mm plastic nut that rounds off very easily. I use an 11mm socket to loosen/tighten the valve then continue to open by hand until fluid runs. Wait for all air to be evacuated from the system (2-5 min... maybe 10-12oz of fluid pumped through. When done, vent the pressure by opening the top of the motive bleeder and remove from the car, being careful to clean up any brake fluid drips immediately as it will eat your paint!
8. Supposedly this will be enough to bleed the system, but I usually have to do a second bleed the '2-person' way. Person #1 opens the valve then yells "stomp and hold" to person #2 who stomps on the clutch pedal. It will go straight to the floor and stay there. Person #1 then closes the valve and yells "lift slowly," where person #2 lifts the pedal by hand (or using a little rope if feeling lazy) slowly (10 seconds to lift all the way up). You repeat this process 6-10 times until you can't hear any air bubbles rush through the system when 'stomping.' You shouldn't see any bubbles coming through the bleeding hose either. When done, tighten up the bleed valve and cap it with the factory supplied rubber cap. Throughout this entire process, stop occasionally to check the brake fluid tank to make sure there is still plenty of fluid.
9. Unwind the tape from the slave cyl. This was keeping fluid out of the slave cyl. up to this point, but we need to get fluid in there now, so unwrap it, allowing fluid to fill slowly (5-10 sec, if possible)
10. Mount the new slave in the bracket and tighten up the 2 bolts.
11. Climb into the car carefully and pump the clutch pedal 3-4 times. Make sure your clutch pedal springs back and that you can shift easily into all the gears with the clutch pedal on the floor (engine off). If you can't, repeat the bleeding process (at least step 8, if not steps 7 and 8) until everything works.
12. Examine your work for leaks (after pumping on the clutch pedal a bit, you should see leaks if there are any) and peek under the rubber boot for leaks too (in case your new slave is faulty).
If you want to bleed your clutch again after the clutch slave has been installed in the car, you need to compress the slave again, but this time the duct tape will be hard to use. You will get a slightly better bleed from a well duct-taped cylinder (or using the proper MINI tool), but this will do fine. I generally get a better bleed with my crude tools than when I've had factory mechanics do the job because I'm willing to take the extra time to repeat the bleeding steps a couple of extra times where they would have said 'good enough' and moved on. The more air you get out of the line, the better that clutch will feel.
I cut a bit of a scrap 2x4 down to hold it in a compressed state. MINI makes a nice tool to do this, but I had the 2x4 and a saw handy. There is a slight angle to the cuts on the 2x4 so it can be tapped into place and really tighten things up nicely.
Note: The fluid line that leads to the clutch slave has a black washer / sealer gasket on it. You can see this connector at the bottom end of the pipe #13 in the diagram, below. Sometimes this bit will wear out, causing a leak at the input connection and not under the rubber boot. The washer/sealer/gasket thing isn't sold separately, but comes with the input line (about 1.5 foot long, reaching up just near the driver's side supercharger horn. You have to remove the air intake pipe and some other bits to get to the top of this pipe, but I have read of some folks that always replace this piece (under $20 at the MINI parts desk) with every slave replacement, just to be safe. I've replaced it once and it took another 30-45 minutes to do, but while you are there and bleeding the system, it might be worth it to replace this part as well (part #13, in the diagram below - MINI P/N 21526759854).
UPDATE: I've had several people ask me about the dimensions of the wooden block that I'm using in the bottom picture to keep the cylinder compressed... It's a scrap piece of 2x4 that I had laying around and I cut it very roughly with a hand saw to make a bit of a wedge that I could drive into place. It might not be the best wedge in the business, but it does a nice job of things for me. A pic of the same piece of wood (marked up with a Sharpie to help me re-orient it and to make sure that I don't throw it away!) is below, with dimensions. You just have to make the 2 angled cuts... the rest of the block is the standard dimension of a 2x4 (which isn't 2" by 4"... go figure!)