Wait. Skyrim is a steam cloud game, how did you lose your saves?
No it's not, if you run it on a second machine none of your saves are on cloud, they are all local. My save folder is 1.2gb and you should find it in C:\Users\(you)\Documents\MyGames\Skyrim
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You may have a write error when you do your reinstall, as the Steam folder is protected. To get around it you can just make install a new steam folder beside the old one and copy everything (well at least the big folders) into it. That way you won't have to go through the hardship of working out how to inherit a folder and finding out it didn't work.