Let's try a standalone Apple Application Support install. It still might not install, but fingers crossed any error messages will give us a better idea of the underlying cause of the issue.
Download and save a copy of the iTunesSetup.exe (or iTunes64setup.exe) installer file to your hard drive:
Download and install the free trial version of WinRAR:
Right-click the iTunesSetup.exe (or iTunes64Setup.exe), and select "Extract to iTunesSetup" (or "Extract to iTunes64Setup"). WinRAR will expand the contents of the file into a folder called "iTunesSetup" (or "iTunes64Setup").
Go into the folder and doubleclick the AppleApplicationSupport.msi to do a standalone AAS install.
Does it install properly for you?
If instead you get an error message during the install, let us know what it says. (Precise text, please.)