How do I repair a 1999 Mercedes Benz s600 odometer light?

The 99 600S is the 140 Mercedes chassis.

I have not worked on the 12 cylinder (which is the 600), but am familiar with the 500 (8 cylinder). You won't like the answer. The light is not serviceable.

On the two 500S I am familiar with, the light is not reparable without dismantling the entire odometer and ambient temperature module. Mercedes sells the modules as a unit and completely sealed and not serviceable, for a very expensive price. However, there are companies which do sell refurbished units, and while a fraction of the Mercedes retail price, they are still hundreds of times more expensive than the light itself. Check the MBUSA owners club magazine - The Star - for some possible suppliers for rebuilt modules.