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 QuickTimeInstaller.exe installer file to your hard drive:
Download and install the free trial version of WinRAR:
Right-click the QuickTimeInstaller.exe, and select "Extract to QuickTimeInstaller". WinRAR will expand the contents of the file into a folder called "QuickTimeInstaller".
Go into the folder and doubleclick the AppleApplicationSupport.msi to do a standalone AAS install.
Does it install properly for you? If so, does the QuickTime Player launch properly now?
If instead you get an error message during the install, let us know what it says. (Precise text, please.)