DIAGNOSIS The LED itself rarely burns out. The issue is almost always a broken wire or a bad connection where the LED board attaches inside the drill's nose.
FIX You need to open the drill's clamshell housing. Once open, trace the two thin wires from the LED assembly back to the main board. Look for a wire that's pulled free or a cold solder joint on the small LED circuit board. Resolder any broken connection.
IF THAT DIDN'T WORK The LED is a simple component soldered to a tiny board. If it's truly dead (unlikely), you can source a generic 3V white LED of similar size and solder it in place, observing polarity.
HEADS UP Be extremely careful with the thin wires when reassembling. They are easily pinched or severed by the housing, causing the exact problem you just fixed.