Update... I wound up buying another iPad (iPad 2). The first time I was charging up the "new" iPad, I had a thought - what if I tried charging the old iPad with a hot charger. Guess what? It worked. The old iPad came back to life almost immediately - and with 100 percent power. It's been working pretty well since.
Recall that the problem had started when I had left the old one charging. I wonder if I had overcharged it. Yes, I know.. supposedly, you can't overcharge, but what if I had. So, now I unplug the iPad soon after it reaches 100%.