non working digit on a single display that moves with the display would first be verifying the digit enable connector pin on the display board itself is not cracked.
If the display connector is OK. check the 100k resistor for that digit. If it is burned open i'd replace the resistor(with half watt 100k), the digit driver and level shifter for that digit. Typical fail mode for this I think goes like this.... level shifter fails locking on the digit bright. This makes too much current go through the 100k resistor and it burns open. Once the 100k resistor goes open the digit is out.
Also can have a display glass itself go bad. I have some glass where a single digit refuses to light up. I find it in stern 7 digit glass the most often for whatever reason. I'd exhaust other stuff before declaring the glass bad.
I hate probing around the high voltage, so as long as you have skills to solder on the board... this is a case where i'd whack-a-mole / shotgun replace part.