When you removed the old version of Flash, did you use the Flash removal tool? You shouldn't just use Add/Remove programs on your PC to remove flash, otherwise the removal is incomplete and may explain why you are experiencing difficulty installing the latest version. Go here:
http://kb2.adobe.com/cps/141/tn_14157.html
If you haven't already, just download the version for Windows on that page, and run the program (which is very small). This should properly complete the uninstall for you. Then try installing version 10 again and see if you have any luck. If not, post back.