Just a few hours at the Apple Store is all it took. By yesterday evening, the same day that we dropped the iMac off at the Apple Store, we had a message on our phone that it was fixed. They had to replace the motherboard, or midplane as they called it. Now, not only does the computer work, but also the Bluetooth module is fully functional again (or so it seems so far).
It was all under warranty, but this experience has made purchasing the extended warranty look pretty attractive. It would extend the 1-year warranty to a 3-year warranty for about $160. We'll likely get that insurance sometime within the next few weeks.