REVISED May 2013The repair, particularly if you just replace capacitors will not last very long unless you improve the cooling.
See the infra-red camera pictures showing clearly the effect of the fan on and off.
Infra-Red pictures of Time Capsule
Apple built the TC with a fan, but it is not used in normal operation. Only if the TC detects overheating, it will turn on the fan and take the TC offline, probably shut down the hard drive. So in other words, the fan is useless. The TC depends purely on natural convection... and the reason it is dead now is lack of large enough ventilation holes and insufficient care in the design for natural convection which is much harder btw than using a fan.
This is a fairly major mod... I think the alternatives are using an external power supply to offload some of the heat generated or fan mod. Simply repairing the power supply and keeping it internal will not improve the situation one iota. (as a rough calculation I would say hdd and power supply each contribute about 30% to the heat load, and the main board -- router - wireless - controllers about 40%.. but it is the power supply that is the most fragile. Hard drive and board components are pretty rugged.)
So to do this you need to remove the fan (the rubber anti-vibration mounts can be easily pushed through the holes with any round blunt rod about 2mm dia.) and drill or punch a hole through the baseplate under the fan.
Here is my new method..
Pull out the fan from the TC.. You might need to pull the tape back. You can also cut off the rubber mounts if you want to invert the fan.
Snip the 5v wire.. far one in the picture and cut a small piece out of the second one.
Exactly as per picture as it is easier to see it.
The actual wires are numbered on the J1 socket..
1, 5v socket has small white arrow. Far wire in the picture.
3. Control. We need to cut to turn the fan on.
Our control components.. one 3mm green LED.
One small NTC 100ohm thermistor..
Wrap the leads of the thermistor around the LED which is stiff plated steel
Solder them together..
Note the long lead is + .. there is no polarity on the Thermistor.
Note2. The older fans may need more start voltage. If so use a red or yellow LED.
Cut a V into the front of the fan. You should be careful using cutters as the plastic will probably crack.
Cut with small saw would be better or grind it out.. messy though. To me the crack is no biggie.. we are going to fill it anyway.
The next one I did I used the large barrel of a really horrible old soldering iron I still use because it is 60W.. I hate reshaping plastic
but you know a soldering iron is a great tool for doing it.. the stink and the tip never the same again.
Push the leads of the LED under the foam.
The long lead of LED is plus and goes to the socket end. Short one to the fan hub end.
Remove carefully a short piece of insulation and solder the wires to the LED legs.
Now plug it back into the TC to make sure it works.
If the fan does not start up that is not an issue.. it will depend on the exact voltage drop of the LED.
I used two different ones, with one it started and the other it didn't
The fan LED will now flash in slow blinking as the control electronics of the fan tries to turn on.
I fixed it by adding a small 0.1uF chip capacitor between the leads.. please expand the picture for the details.
Just a closer shot so you can see the cap soldered in. I pinched it off an old board from anything.. TV.. computer MOBO.
A second one I did required more voltage to the fan to get it started.. as the fan ages this may be a problem with all of them.
A second chip resistor would fix it.. maybe 50ohms but you will need to trial it. Or change over to lower drop LED.
Each color has a different voltage drop, green are fairly high.. red should be a fair bit less.. but the fan may run too fast.
Here is the red LED version.. fan runs a fair bit faster but maybe necessary if you have older sticky fan.
You should test both I think and may need to change it at some later point as the fans do get old and sticky.
Silicon putty your modifications.. so nothing shorts and everything is held in place.. including any new cracks you added!!
Now you can mount it back in the TC once the silicon dries... usually a few hours.. You can even run silicon bead right around the top. Use screws to hold it in lightly until the silicon sets.. you can then pull the screws out. You have a rubber mounted drive again.
Just wonderful stuff silicon.. buy good quality white bathroom or outside stuff. And a gun. You will never regret using it. Later if you need to fix something you can peel it off.
We used to use a 33 or 39ohm 1/2 watt resistor and that is still fine.
This is designed to give a bit of speed control as the TC heats up the fan speed will increase as the thermistor value drops.. which will also drop the brightness of the LED.
I have shown how to drill out the holes under the fan in the aluminium base.. and I think you should still do that. Open just those holes that are immediately clear to the fan.
The picture is in the kits section.
Old method of repair.. directly from 5v on the power supply.