So, good news and bad. Learned a bunch of stuff, but one thing I learned is that I killed the chip in the right shoe. Dang it. However! The left shoe now works yay!

I decided, for the left shoe, to NOT replace the battery until after I re-attached the LEDs and gave it a try. Worked right away! No smashing up the protective case to get at the old battery. That is very likely when I killed the other shoe. Or when I had a short.... 

So anyhoo, one shoe, whoo whoo.  I got the light's flashing, and glued the sole shut with Shoe Goo, so we'll see how well that sets in a day or two.

Now to buy some new innards for the other shoe...