I have a 2 step process for you
1 - Go to the adobe site and download the flash uninstaller
Run the uninstaller, so that you remove all signs of flash from the pc. The reason you cant reinstall is because there is enough of the program left in the registry for flash to think its already there. Once you run the program, REBOOT!
2 - Go to the Ninite website.
In the Runtimes section (little less than half way down), select flash and install it from there. After the install, REBOOT.
I use Ninite to update and install a lot of programs because its the current version, they don't install any bloatware with the install, and you can install multiple programs from the list at one time....
If you need any other help let me know.
Head of Tech Geek and More