
I do not know if every indiglo watch work the same way, but I recently cleaned up one where the light did not come on.

Try carefully cleaning all of the contacts including the presser for the light with contact cleaner or alcohol and see if that helps. Some glowing watches use electroluminescence, and this also requires a voltage boost, and a relatively high current to make it work. The circuitry needs a relatively high current, typically 10 to 20mA compared with the tenths or hundredths of a mA, or even a few uA needed to power the movement of the watch.

LEDs do fail, but typically last several orders of magnitude longer than old fashioned filament bulbs, but white LEDs require a minimum voltage before they will light, somewhere of the order of 3.3 Volts, and in a watch that uses a 1.5V battery, this is produced by a boost circuit. The watch movement would be less likely to be affected by this as it draws very little current, but a higher resistance in the contacts might be enough to stop the light from glowing.

There is a possibility that the battery contacts are not very clean.
