I'd start by verifying power supply to the IC's on that board and also scoping the power to see if it's stable. If that looks good, figure out how the Zilog CPU is connected to the LEDs and the 7-segment displays and see if anything is happening along that path. At a glance, I'd say the CPU is connected more or less directly to the outputs & inputs on that board with not a whole lot in-between. Maybe some of the 74-series IC's are interfacing between some of the LEDs and the IO ports.
If this were my machine and it would be important enough to try and salvage it, I'd lock myself up for a couple of days with a stack of datasheets and try and draft a model of that board to see how everything relates to each other. Then form a couple of hypotheses on what might be wrong, do some measurements to verify, etc. It's a time-consuming process that would be accelerated dramatically if a service manual or at least a schematic of the board would be available.
Checking for visible damage is a good place to start, but of course not guaranteed to solve this.