I’ll first assume you have explored any sleep mode settings that might be available, and disabled them or set them for a long period.
Open it up. Check connections on the power supply and output voltages on the power supply.
Run it until it goes dark again and re-check those voltages.
If they are the same as before (i.e. non-zero) then it’s not the power supply.
If some of the power levels have flatlined then that’s your culprit.
The fact that the power light stays on is weird and suggests that if it is a problem in the power supply it’s only partially fried. But that is entirely possible as it probably puts out multiple voltages independently.
Since the sound goes out too, that suggests it’s not just the power to the backlight (cold-cathode flourescent tubes probably have their own distinct part of the power supply output).
There could be a lose connection between the input and all the video and audio output sections. Are there alternative inputs you can put the signal into?
Can you try another source (e.g. a laptop? old VCR? stand-alone DVD player?) and see if it behaves the same or differently ?