It's very likely that the component soldered in at that spot failed or shorted, and it is possible that it burned up before doing any further damage. You'll need to find a schematic diagram of that circuit board to determine exactly what goes there, and then source the correct component. Unsoldering the remains of the old one, cleaning the contact points, and soldering in the new one should be relatively simple and straightforward, and provided no other components were damaged by the overheating, it should be good as new.
Back in the late '80s and early '90s, I had two different Craftsman rechargeable screwdrivers fail in this same way. Something on the little board where the charger plugged in burned up. In both cases, it was exactly the same component. This was pre-Internet and I had no way to determine what that component was. In both cases, the tool worked until the battery ran out, and then there was no way to charge it. I chalked it up to a design defect, and made sure after that to buy cordless drills and screwdrivers with removable batteries and separate chargers. This was also where my firmly held theory of circuit boards as self-destruct devices had its origin.