In the control panel, under "Sounds and Audio Devices", it showed No Audio Device, but going to the driver, it showed it was working fine.
Online searching finally led me to the "Administrative Tools|Services" tab. Windows Audio was disabled. I set it to automatic and started the service, and all is fine now.
So I have a brand new soundcard that I really didn't need after all. Then again, I didn't have to completely wipe and reinstall Windows.