The "for older video cards" version will probably work for you judging from other threads I've been involved in.
You can try a full tear down and then use the individual .msi files as you have done before. Did you get iTunes 12.1 installed before trying 12.1.1? These last two builds are the first full 64-bit versions of iTunes, previous 64-bit releases (including 12.0) have been 64-bit installers for mainly 32-bit code, particularly the main executable.The "for older video cards" version is similar to previous releases which may be why it works when the other versions won't.
Or you could try logging the installation to see if you can spot any errors that you can correct. Install AppleApplicationSupport.msi and, if relevant, AppleApplicationSupport64.msi first. The following commands log the install of AMDS for 32-bit, 64-bit or the new iTunes6464setup.exe versions. They assume the .msi is at the root of drive C though you could edit accordingly. If you want to type out the command the option switch in the middle is slash el star vee for verbose logging.
msiexec /i C:\AppleMobileDeviceSupport.msi /l*v %USERPROFILE%\Desktop\Log_AMDS.txt
msiexec /i C:\AppleMobileDeviceSupport64.msi /l*v %USERPROFILE%\Desktop\Log_AMDS64.txt
msiexec /i C:\AppleMobileDeviceSupport6464.msi /l*v %USERPROFILE%\Desktop\Log_AMDS6464.txt
You can triple-click a line to select all of it, then press Ctrl+C to copy, switch to the command box and click the icon at the top left of the window to access the paste menu.
The log files might provide further clues as to what is going wrong. I suggest using Notepad++ to review the logs. Look for words like "failed" in the log.
tt2