I have replaced the fan in my X200 before and it will require you to remove the motherboard. I say if you can do this then we will ship a badge indicating you are one of the 'Thinkpad Elite'
Is your fan not spinning at all??? If it is spinning but your X200 shuts down you might want to get access to view the innards of your X200. It could just be enough gunk in the fan assembly and exhaust that is causing high temps. If the fan is truly RIP and if you are receptive to:
1) Setting aside an afternoon for your repair project
2) taking a deep breath
3) getting an uncluttered workspace ready
4) downloading the Hardware Maintenance Manual for your X200 (you get this from Lenovo's website)
5) good lighting
Then fan replacement should be possible.
HERE is a thread that might help. 'Turqoisegirl08' is yours truly and in the thread I helped guide a forum member at NBR through their fan replacement. The person really was nervous and afraid but took initiative and dove right in. The end result was the replacement of a noisy fan on its last legs to installation of a fan that ran silently.
For my own fan replacement for my X200 it took ~3.5 hours. I made sure I had clean white printing paper to draw maps of my X200 and pushed the screws thru the paper at corresponding locations with the X200. I did this so I would not mix the screws up and cause myself some frustration if for some reason the screws got mixed up.
You will also need some thermal paste. If you do go the route of a replacement I would suggest looking for a fan that is Lenovo branded. I was lucky with mine as I scouted out fleabay and found one that was unused and still in the Lenovo box. I do not remember exactly how much I had paid for it.....perhaps ~$28.
The only way you will be able to identify what type of fan your X200 is to take the keyboard off and look at the FRU of the fan (fan should be on the upper left side of your X200). Write all the numbers from the fan assembly down on a notepad or a scratch paper. There are two or three different FRU of fans and fan assemblies for the X200. I suggest buying the whole fan assembly which comes with the copper heatsink. I also suggest getting rid of the thermal paste that is already included on the plate that will make contact with the CPU and applying your own. The paste on mine was very thick and surely would have made a mess when I got the fan assembly on. The thicker the paste the less effective the heat dissipation so you need to make sure you have a thin even layer across the CPU.
Good luck!
