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Does any of the other backlights ( the lights that illumate at night)work? For example, the lights on the instrument cluster, the backs lights on the heated seat switched, the backlight in the hazard switch? Also does the climate control stick work? you still have and are able to control heat and air conditioning?
Ok, the light that illuminates the climate control unit are on the same circuit as the lights that illuminate your instrument cluster, the hazard light switch, the seat heater switches, and for the most part the lights that illuminate any other buttons or switches on the dash that allow you to see at night. So if your able, turn on the headlight switch, this will turn on all the back lights as well. See if can see any other lights on. I would look at the hazard button, you may have to cup your hand over the button to block out all other light coming in from the car, then look closely to see if the button is illuminated.
If the button is illuminated, then problem lies in the climate control unit itself or less likely is the wiring leading up to the unit. Please let me know your results.