On my 1995 the only light that dims in response to the headlights being turned on is the radio display. Additionally, the radio buttons and the cluster illuminates. Obviously you can continuously vary the brightness of the cluster lights, thus they cannot be traditional LED bulbs. I'm about positive digital radio readouts use LEDs though, at least they are refered to as LEDs, and they do in fact dim.
The Climate Control button lights, however; do not dim in response to the headlights being turned on. They are illuminated with the same brightness no matter what, obviously so you can tell which button is selected in the daylight. Additionally, when the headlights are turned on, back lighting is added to temperature and fan speed selectors. However, as this is a 1997, I expect the temperature and fan speed readout is a digital display that, like the radio, dims in response to the headlights.
However, I believe the question is directed specifically at the indicator lights contained under the function selector buttons (i.e. Heat, A/c, Vent, Defrost...). This could be surface mount LEDs, LED bulbs, or any other PCB mounted bulb.
Repairing it at home will require taking it apart and knowing how to solder on a circuit board. At the very least, the bulbs will need to be replaced.
I agree 100% with pcmos, the fact one entire row of lights has stopped working, it is most likely a bigger problem than burnt out bulbs. The power has been cut to the row for some reason... Most likely because the PCB (circuit board) has been cracked.
Have the buttons ever been pressed/pounded/hit harder than they should of been?
Most likely a wiser solution to buy a used climate control console and just swap out the circuit boards. That's the easiest, the most cost effective... but not the cheapest. (Which is always my first concern. LOL)