Well I resolved the problem, but here's the details for reference.
When plugging in the USB mouse, instead of automatically installing like normally, Windows would display the driver install dialog, etc. which would ultimately result in a "data is invalid" error (whatever that means).
Device Manager would show that the drivers were not installed, but was otherwise detecting the mouse.
I finally checked Device Manager on another PC to see which driver files were being used for the mouse and "hid-compliant" devices and copied those files (found in system32 and system32\drivers) to the problem PC, overwriting the originals and that fixed everything.