The advice so far is very good.
1) Boot into your motherboard BIOS (DEL key usually) and ensure that the onboard HD audio is enabled, then save.
2) Go into Windows, and install (doesn't hurt to reinstall) the following drivers/software:
a) Main Chipset (see motherboard support site)
b) Audio software (see motherboard support site)
3) Go back to the Audio->Playback Devices link->.. and find the onboard solution (probably Realtek) and switch to that.
Other:
1) As said, if still stuck see if errors in Device Manager:
http://www.computerhope.com/issues/ch000833.htm
2) If your HDTV has a headphone jack you MIGHT be able to use that. If so that's better if still using the TV as you can just plug in the headphones and not worry about manually switching via software.