I'll reply to myself again. Woohoo!

I completely uninstalled every video codec I could find on my PC. I had to manually delete system files and registry keys relating to DivX and XviD. I eventually got it to the point where WMP would open the Matroska file, but no video or audio would be present, and the properties page listed no codecs in use at all.

I then installed DivX from their own site, and followed that with the Nimo codec pack (with DivX de-selected).

The problem is still there just like it was before.