Does the glove compartment light work? If so (and I think it only works with the key on), then the problem is the lighter socket itself, or the wire has disconnected from it up in the dash. This can be a real problem to fix because the console pieces in our cars, if not already in poor shape due to age, are terribly fragile and don't like being taken in and out repeatedly without cracking.

As for the illumination of the climate control - that's a bulb issue, not related to the cigar lighter.

Report back on whether the glove box light works - that's on the same circuit as the cigar lighter.