Hello, my name is Dan.
I will do my best to help you, the only stupid question is the unasked one.
One fix is to run a separate audio cable from the headphone port to the TV, inelegant but effective.
Another is to restart and perform a pram reset, this resets all the USB and outputs on the Mac thus perhaps letting it 'see' the TV.
To do a PRAM reset, you need to restart your machine and hold down these 4 keys all together at the same time.
‘Option’, ‘Command (Cloverleaf or Apple)’, ‘R’ and 'P' all together at the same time.
Keep them down until you hear 4 startup chimes, this is the sound that your computer makes when it starts up, it is like a boonngg. The screen should also cycle from white to black and then back to white along with the chimes.
The sound should also get louder.
This resets all of the system buses on your computer, USB, FireWire, display, ethernet and your keyboard/mouse/trackpad.
This will correct many faults but not necessarily yours.
Try that and let me know.