I tried a different cable, as suggested by Cougar-Duck, but that wasn't the problem.

I mentioned earlier that I could navigate to the Ipod through Windows Explorer, and see it just like another hard drive. I was able to format the Ipod, using the "Restore default settings" option.

Once formatted, I started up ITunes, and it saw the device, and asked if I wanted to restore it. I clicked "yes" on a couple of popup messages, then ITunes reinstalled the software. The Ipod is once again working great, and is now playing our Christmas playlist on the speaker system in the front room.