Last update:
The problem has been solved! Took a bit of doing, and isn't exactly what I would recommend in all cases, but it did work for me.
I took out the hard drive (SATA) and attached as the boot drive in another SATA capable system with PS/2 ports for the keyboard and mouse. When it booted, I ignored all "found new hardware" messages for anything that wasn't directly related to the USB ports and peripherals. Bingo! We have a keyboard and mouse now! I went into Device Manager to make sure that everything was as kosher as it could be for the keyboard and mouse then powered down the system and installed the hard drive back in the original system.
When the Windows booted up, I had a keyboard and mouse again and was able to finish working on the spyware infections that probably caused this problem in the first place.
Thank you for all of your help, guys.