Hi RippyD, let me try to help you with your issue.
It would also be helpful to know the BIOS version you have and if you see any bangs in device manager. Just in case, try connecting the Intel® NUC directly to the TV.
The two models you indicate above are basically the same board; it is just that the D33217GKE is sold as a bare board and the DC3217IYE is sold as a kit, with the chassis.
My BIOS version was recently updated to 2014.0522.1323. I do have the chipset drivers installed - version 22.214.171.1247 per the Intel driver update utility. For my audio driver it says "Intel(R) Display Audio. The driver version is 126.96.36.19937. The note below this says "This device is unknown or unsupported. Please contact the manufacturer for possible updates."
It's a little disconcerting that the Intel driver utility concurrently says that I have the Intel driver installed and also that the device is unknown or unsupported. How is this possible? Is this an indication of the issues I'm having?
The tool is not properly recognizing the audio driver version you have, which is indeed the latest one.
Did you verify the audio is enabled in BIOS (Devices, Onboard Devices)?
I don't see any audio device in the BIOS. Under Onboard Devices it shows LAN under Onboard Device Config and Num Lock under Legacy Device Config.
My bad, I realized I took a different model to check the BIOS options.
So, I think you should try testing the Intel® NUC with a different TV and cable, hopefully a straight-thru cable and not an adapter. You may also run a BIOS recovery or contact your nearest Intel support center for further assistance.