before you get too far.
if the dimmer switch is dimming the odometer display, it is working as there is only one feed to the instrument cluster (based on my 01 FSM)
since this means that the prob is in the instrument cluster, you could hot wire 12 volts to the feed coming out of the cluster to the hvac controls. it would also back feed into the cluster and everything may light up. this is risky but meh, the cluster would be considered crap anyway at this point.
'01 TJ Sport (black), 4.0L, 5speed, Dana44, Flowmaster 50, 2"BB + OME, 32"ATs on 16" alloys.
'01 Suzuki GSX-R750(red), -1T 520 kit.