Oh dear... another problem there then...I'm afraid it'd be listed in that, which you can't access
Okay, go to SafeMode (del at bootup) and try going to it from there. If this lets you open EXE's in the normal fashion then this should work.
So enter the Add/Remove programs and choose Windows Components and then delete it (EDIT: Oh...its not listed there
Okay, hard delete the program (its stored in a different place on this machine, though should be in c:\Windows i'd think.
If not, run Windows Search, and search for files and folders from safe mode, and search for 'wordpad.exe' and then delete the program which is a complex way of doing it, but seems the only way, and if it fixes your problem, you might as well.
Its weird though its choosing programs to work with and others not too... that makes me think its a bug, though the online scanners didn't find anything.
I hope this suggestion helps, and it's my last straw apart from a full re-install i'm afriad, so I hope it fixes it