Mac mini sleep LED fades on once and won't boot
Based on what you described, you most likely have a logic board issue.

The SIL (Sleep Indicator Light) will normally come on and stay on until the GPU is initialized at which point the display turns on.

From Apple Support:


  When you press the power button to turn on the computer, the SIL will also turn on. As the computer starts up, the SIL stays on until the processor (GPU) has initialized and the display turns on. When the GPU initializes properly, the display will turn on and the SIL will turn off simultaneously. 


If you have a problem, the SIL will blink out a "code" which you can use for diagnostics.  Since it's shutting down, prior to an error code or a boot, the most likely cause is your logic board.

It's time to take it in for service.