Water damage, boots, runs, battery won't charge

First thing I would do is inspect the circuit for signs of component damage.

Bear in mind that we're working with very small components and often multi-layer circuit boards on this kind of device, so the damage may not be as obvious as you might think such as a whopping great black scorch mark across half the board.

Start with a quick visual scan of all the larger components looking for any signs of discoloration particularly around the legs of chips and capacitors. Sometimes you will see a small crack in a chip where the internal circuit has shorted and cause the case to split.

Sniff the circuit board (yes, get your nose in there) around any capacitors. If they have blown they give off a distinct very strong smell, I don't need to describe it because you will know it if it's there.

Then look for signs of corrosion around the metal legs of components, white powdery residue is a good sign of where unwelcome moisture has been.

If the board is not too heavily populated with surface components, hold it over a bright light and look through the circuit board itself. If it's a multi-layer board, there will be tracks running through the sandwiched layers of the board. These won't have come into contact with the moisture, but those tracks tend to be thinner than the surface tracks and if a short elsewhere has caused more current than usual to run through them, then it could be those buried tracks that burn out. You will see dark patches around the damaged areas.

If you manage to identify any of the above, then that's the best place to start your investigation into the components. Larger surface mount components can be easily replaced, some chips can be easily bought (particularly generic ones used in power circuits), but once you get down to the smaller components, then you will find it harder to fix them.

Hope that helps to narrow things down a bit for you.