Thanks for the replies.
There was no response at all. I wound up buying 2 new batteries and an external charger, and also a new OEM charging cable/USB adapter for the phone, all in all it cost me almost a hundred dollars.
The old battery charged in the external charger, I then put it back into the phone.
I had used the charging cable to charge another smart-phone once, and wondered if I fried it somehow. I did notice the original cable has a big barrel on it (I think it's a surge protector or something?). My other phones don't have those.
I still don't know the cause but the external charger worked on the battery.
Edited to add:
I do remember reading not to power the phone ON while charging. If I remember correctly it said ON or OFF but don't change status while charging. I had warned my mom about that.
I'll remind her of it. But when I got the phone to "fix," it was completely dead, so I have no idea whether it was on or off. I don't think she changed the state, however. But if the battery went dead before the phone was powered down, maybe it caused the problem? Seems kind of fussy, if that's the case.