Do a search using keywords 'panel light rheostat'. I don't know if your rheostat is like the one on my '85 380SE, but regardless, they all work the same. If you can get the instrument panel out, just jumper a wire across the terminals into and out of the switch, as if the rheostat was turned all the way 'up'. If the lights come on, then you can either fix or replace it, or solder a permanent jumper in its place to illuminate the dash/lights at 'full' brightness (which isn't very bright anyway, so this thing is, IMO, useless).

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1986 560SL
2002 Toyota Camry
1993 Lexus