DIAGNOSIS Either the LED itself has burned out (less common) or, more likely, the wire connecting it to the board has come loose or broken, often from vibration or a previous drop.
FIX You can do a quick visual check. 1) Look into the light port near the chuck. Is the tiny LED lens visibly cracked or damaged? 2) In a dark area, pull the trigger and look very closely at the light port. Sometimes a very faint glow means it's getting low power from a bad connection. If you see nothing, the next step is internal. The light is part of a small wire harness that plugs into the main board. Fixing it requires opening the housing to check that connection.
IF THAT DIDN'T WORK If you open it up and the connection is secure, the LED module is a simple plug-in part you can order. It's not soldered. Unplug the old one and plug in the new one. Before you order the part, use a multimeter to check for voltage at the connector when the trigger is pulled to confirm the board is sending power.
HEADS UP The worklight is designed to turn off shortly after you release the trigger to save battery. It's not supposed to stay on. So if your problem is "it turns off immediately," that's normal. The issue here is it never turns on when the trigger is pulled. Also, the light isn't a critical function for operation, so you can keep using the drill safely if you can live without it.