If your audio device completely vanished from the Windows Device Manager list, the onboard audio chip might be turned off at the hardware layer.

Download the SoundMAX Integrated Digital Audio driver for Windows 7, 8, 8.1, and 10 (64-bit) for LAM - Core2 Duo 2.2 - DriverIdentifier

Integrated audio chips (usually made by Realtek, VIA, or Conexant) soldered onto the motherboard.