It looks like the only one listed is the Intel Display Audio and this would be for the HDMI port. It does appear that the actual sound card is missing or may have failed. I do see there is one Unknown Device and I would like to find out what it is to verify if it is related or not.
Open the properties of the unknown device and navigate to the details tab.
Change the property to hardware IDs and report them to this thread.
I'll keep an eye out for your reply. :)