Look at the main board. Does anything look wrong or discolored? Maybe there's a problem there. In my case, there was corrosion from water damage in various places. But what was really damaged?

Even though the battery voltage measured a reasonable 3.6 volts, it was dead. Temporarily wiring in a fully charged battery (you do have spare lithium ion batteries around, don't you?) revealed one particularly hot component. Yes, be careful when touching components on the main board, and look and smell for telltale signs of smoke or overheating.

Once you figure which component may be bad, remove and replace it. If you don't have any replacement, still remove it and test just to see how well it can limp by without it.

In my case, I removed one component, but it was actually an adjacent capacitor that was getting really hot. Removing that capacitor brought some functionality back- the LEDs actually lit. Replacing the original component and putting in a new capacitor brought everything back again.