I have had good luck getting help on the 8th (2008 Civic) and 9th gen (2012 Civic) Forums in the past, surprised they were no help in this case.
Best I came up with was this:
https://www.clubcivic.com/forum/thre...-civics.59115/
But since you removed the cluster already and say it is an LED, not sure what to make of it except that perhaps the CEL is LED and doesn't normally go bad.
Saw another post that said check for a fuse (#43 I think) but seems to me that fuse would power more than the CEL.
As you have access to it, can you put power to the LED to see if it lights?
If it does, at least then you'll know to trace the circuit back from there.
Would be nice if you can fix it for cheap so that the crook doesn't win in the end.
Good luck.