Some good and OK news on my repair processes:
My old unit (killed by measuring AC and some ground connection via USB to earth mains -- from my LCD replacement solution) I finally got some life back into it. The ADC and integrator basically had no activity. It turned out to be one of the traces on the board for U105.
I had already replaced U105 (it was one of the chips which died), and I added a socket. But either the trace was vaporized by the original problem, or bad soldering work. In any case, no matter how many times I tried to reflow the joint to include the trace (the cut in the trace was only a millimeter or two away from the U105 pin that it connected to), it wouldn't connect. So I gave up and just used a short bodge wire.
After that, now the ADC is running again!
That older meter is not out of the woods yet, though. The only range that is even close to accurate is the two lowest voltage ranges. All resistance ranges and all but the lowest current ranges show basically nonsense values.
I'm also suspicious of the stability of the voltage measurements. I used my Keithley power supply (2304), which isn't the cleanest supply (although the noise is very ... random? gaussian?). And basically no matter what setting I pick for the Keithley, the meter instantly locks on to some (reasonable, if a little bit off) voltage value -- all 5.5 digits, no drift at all.
This makes me very suspicious, but it is certainly progress compared to before.
As for the newer of my two meters, I have been testing by completely disconnecting the negative voltage rail entirely (by removing the resistor and capacitor), and powering it using the aforementioned Kiethley power supply. One of the cool features of the power supply is that it reads back (and sinks) the voltage and current present on the connected nodes, no matter if the supply is ON or OFF. Furthermore, you can "force" the node into compliance.
This came in handy when I noticed that, when the power supply was OFF, the negative rail came in at -0.640V with some small (sub 1mA) current. In other words, the negative rail was going positive somehow. I suspected this was because of some short to a positive rail, but I was unable to find it. In any case, I tried correcting it by forcing the rail to zero (by setting the compliance voltage to 0.001 volts). Because the current measurement on the supply is so good (it has a 5mA range), I could easily see leakage currents change as certain multiplexing JFETs were turned on or off! We're talking values like 0.0145 mA to 0.0165 mA!
In any case, when I powered the negative rail with 9V, things lit up. Again, pretty much all of the measured values are pretty far off; but at least it's alive. And there isn't the kind of stability the old meter has. I also am pretty sure that two of the multiplexing JFETs are bad: Q110 and Q112 both exhibit a pretty huge drift (like, 100s of mV) in the gate voltage after being turned on (that is, when the voltage is near or above 0V). I can't imagine that's correct, and it's slow enough (and bidirectional) that it seems thermal.
I'm sorry for the rambling, I'm just putting a log of what's going on in case it randomly helps someone else. I think I would have made more progress quickly if I had just went ahead and removed the multiplexing JFETs, but I really don't want to unnecessarily contaminate the sensitive/guarded sections of the meter, because if I screw up calibration I really have no way to get back to calibration.
For what it's worth, the military manual that was posted above is a great resource. None of the Keithley 197 manuals so far (there are three or four editions out there) have an actual picture of any waveforms (be they oscilloscope traces or just theoretical diagrams). I knew what the waveform of the 197's integrator/ADC step should look like because of the Keithley 175, but you never know.
In any case, despite the fact that the military version manual uses a different CPU (and a different crystal frequency), all of the troubleshooting values in the military manual are identical to the steps listed in the normal Keithley 197 manuals, except oscilloscope
traces are pictured. I don't know what the "alternate" trace is, but I don't see it on either of my two units.