I tried dozens of times. Basically I tried all sorts of different things from editing registry entries, unregestering the msiexec file and re-registering it (I am not even sure what that's supposed to do), replacing potentially relevant files, virus/adware scan, safe mode installation, making sure all other programs were closed, disc cleanup, installer cleanup, restarting (obviously). I ended up wiping the drive clean and re-installing. Not surprisingly, that did the trick. If that seems drastic, I will mention that I was also having MSI problems with a new printer installation (which may have been what started it all, but may have just been the first sign of trouble).
Anyway, thanks for the advice. Sorry I gave in and didn't try any of it (aside from numerous attempts at installing). I am also sorry that I cannot offer any advise for those who find themselves in the same boat as I was in. But if you do, and if you don't have much installed on Seven already, then you might be better off saving yourself a few hours and just re-install Seven.