Someone helpfully suggested a docking station that would power the computer through a different port while it was being used, and that sounded great--but I can't find one that says it's compatible with a Toshiba Satellite A75-S206. I don't know enough to really know whether there's a dock out there that I could use . . . I really just want to be able to power the thing, the other features don't make any difference to me.
Then I thought about something else--what about a way to charge the battery outside the computer? Surely that's doable, right? Nope. Apparently the Li-ion batteries in these things are excessively 'splody and they're all completely different, so there's no generic charger and a homebuilt unit would be beyond my tech skills.
This leaves me with opening it up and re-soldering the power jack--again--myself. The prospect is frightening, yet strangely exhilarating. And if I completely screw it up, the case is large enough for several Shoot-N-C targets to stick.