Well, I took the broken flashlight to work. I have colleagues there who are genius electrical engineers and one in particular, Sam, had a fun time with this challenge. I explained to him the situation and handed him the circuit board. He peered closely at the board, rotating it round and round as he traced all the connections. As he did so he made notes on my whiteboard and after a few minutes he had deciphered the full schematic of the circuit and it's intended operation.
We then hooked up a bench power supply and took some measurements (with a much nicer multimeter than I have at home). We measured 2.45 mA of current flowing when 6V is connected and the lights are off. At 220 mA hours of battery life:
220/2.45 = 89.79 hours 89.79 hours / 24 hours/day = 3.74 days
Which totally correlates with my experimental data. Here's the schematic for your viewing pleasure (click to make it bigger):
You can see some measurements we took and a curve for the diodes. Yes, it drew a lot more current from the bench supply than it did from batteries. We started with those diode equations but didn't end up going anywhere with them. Sam thought maybe the transistors weren't turning all the way off so he soldered resistors between the gate and drain (or was that the source...it's been a while since I've had to know that) to pull the gate voltage up higher and force it into cut-off, but that didn't help. Then we just measured voltage across R1 (going into the chip U1) and calculated the current through it and it turned out to be 2.45 mA. We probably should have done that first, but oh well. It was fun even if we couldn't fix it right there.
I've been thinking about going to radio shack and looking for another small switch to use. You see, I don't really need three modes. On and off would be fine, so I could just put a new switch in series with the batteries, cut the trace to the chip, and bypass the transistors, just making it a simple battery, switch, resistor, LED circuit. That would be nice and simple and just as effective.