The dash lights are controlled from the headlight switch. It sounds like a wire either worked loose from the switch or became threadbare and shorted, which is where the fire came from. Getting to the switch is not easy on the 54s but can be done with some patience. You can try working yourself up behind the dash and with a flashlight you might be able to see what shorted. I was able to fix a short in my 54 without having to drop the dash. But, if not...read on.
You will need to remove all of the screws that hold the upper and lower halves together. Most all are located under the leading edge of the dash. There are also a few bolts that attach the dash to the underframe, as well as some that hold things like the steering shaft brace in place.
Once you have all of the screws and bolts removed, gently begin working the lower half the dash down from the upper half. Don't try removing the upper half...it's difficult to get at and it won't help with what you're doing. Remember, all of the wiring is cloth covered and 60 years old so it will be brittle. Work the lower half down only as far as is needed to gain access to the switch and wiring. You shouldn't have to remove the switch, I'm sure the issue is with worn wiring.
Make sure the battery is disconnected before you begin!