Mac Mini – No Sound Except for Startup Chime

Odd though it may seem, try launching Garage Band (if you have it) and creating/saving a new project. It seems that Garage Band does some low-level tests on your audio system when it runs, and clears up weird states like you're encountering.

This doesn't answer the question of what's wrong, but hopefully will fix it anyhow.

The red light in your audio out jack indicates that it is active. That is why you do not hear anything on the speakers.

Your analog/digital switch in the port is stuck -- the red light is the digital audio signal.

Try to plug the headphone in, this time wiggling it from side to side while it's plugged in.

Alternatively try cleaning the audio plug:

Try inserting a cocktail stirrer or a toothpick and jiggle it around in the audio output port and the internal speakers should come to life.

If you have a compressed air can use it. that seems to work best.

There is a switch inside that has to be tripped.