First attempt to resurrect the electronics after pulling them out of the right shoe. All work done on my Twitch Livestream, thank you to all the good-humored engineers who helped out. 

It took two tries to remove the old corroded battery and add a new one. I managed to cause a short and kinda fried the first one, oops. Once the new battery was in, the multimeter showed that the little circuit board was still working, though at this point every wire had fatigued off. I also now see how the flash activate! There is a soft spring in a protected compartment that bounces against a metal plate to complete the circuit. Nifty!  The 4 LEDs also work, and turn out two are red and two are yellow.

I glued the spring compartment back together with a touch of Aron Alpha, and secured the new battery to the board a little tighter with hot glue to reduce wiggling. A few other gentle cleanings with alcohol to get ready to start rewiring.