I performed a hard reset - held the power and home buttons down until a black screen appeared with a white tablet and cable - and left it alone, just keeping it plugged in to the wall charger.
THAT screen stayed up about 5 minutes, then rebooted to the white screen with the black apple, the the login screen came up. It had kept the hard reset screen up long enough to bring the battery up to 6%, so I could shut it down to charge all the way.
Ridiculous issue to begin with, but that's what finally worked for me.