I'd like to keep this restoration as faithful to the original as possible, and certainly will in terms of artwork and aesthetics, but I have some questions about the innards that I'm hoping someone with some experience can help with.
Welcome to the forum drjones! I can tell already, I like the cut of your gib.

Short of finding someone willing to part out a functioning cabinet, I'm not sure how to go down that path. I'm contemplating getting a 19" monitor, switching power supply, jamma harness and adapter for the DK boards and going that route, but I'm not sure how feasible that is, either.
You could probably look into using a standard PC monitor with a
video adapter from CGA to VGA. Either that, or try to find a Sanyo 19" on fleabay or perhaps ask here, or on klov's trade forum for anyone selling them. Or...get a different 19" arcade monitor, like a
Weiya or something.
One thing you might want to consider if you can't locate original boards, is the
Braze Double DK board (sold at Mike's arcade), and run it off a switcher. This would give you both DK and DK junior. Though the downside is that the footstep sounds of the original isn't well reproduced on those machines. But if you're desperate, it's not a bad option.
Another route for a board is getting
those 60in1's, and just setting the board to only play DK (you can omit games from the game's list). A 60in1 board has a VGA out, so a standard PC monitor could be used.
Again, this is if you can't locate any original boards.
I'm still trying to convince myself that I'll be able to work with an arcade monitor and not turn myself into a crispy critter. I desperately want to climb the learning curve and not just slap a PC in what is a piece of history.
I can respect this.
My advice is to try to see if any of the arcade trading forums have parts you can buy and just start "piece-mealin'" it together. Other than that, you might have to suffice with non-original internals.
Get your hands on the manual and parts schematics, there should be a full list of components you can refer to. That'd be a good start.
Second to that, do a search on this forum, as well as KLOV's for all DK restoration threads to give you an idea of what you might be faced with. It's good reading to see other people's trials and tribulations.
Also, let's see some pictures of the cabinet!