Thanks everyone.
Corran, the images are indeed on a circular piece of posterboard connected to a knob that turns. My original idea was to use a push-pin thru the front of the console, just above the screen itself to both hold the wheel together and act as the knob, but during construction I found out the friction was causing the push-pin to become separated from the circular piece every time, it would come loose after a few turns and stop working.
After a bit of tinkering, I came up with an alternate mechanism--there is now a much larger knob connected to the back of the wheel, firmly glued into place. The angled section of the computer fits flush to the angled back corner, and the knob connected to the back of the computer sticks out a hole in the outer wall of the playset. The "pin" that holds the wheel in place (and that you can see in the lower part of this picture: )

is now made from an earring I found at the dollar store that looked vaguely like the controls on the computer. I attatched the other earring to the computer too, to make this one look less funky.
No plans on making the elevators work, though I did figure out a way they
could be made to work and retain the ability to come apart level by level (if, say, Hasbro were to devise a real toy based on the design.) For my taste, the tubes were too shallow to try sticking figures in them and making opening doors.
This thing is holding together like a rock--each level simply sits on top of the one below it, but they're so wide it's pretty easy to pick it up by the base and carry it like an enormous birthday cake. The next level should be interesting in that regard, because I'm splitting it down the middle and building it as two dioramas that fit together. The column that will hold the Tractor Beam controls will sit atop the seam, essentially making level four a three-piece construction.