The problem was that the phone wouldn't charge. So I was trying to determine if the battery, charger, or phone was bad.
I took a chance and connected a battery from a different phone using jumper wires to the pins on the phone. I connected positive to the pin corresponding to the + sign on the battery (obvious enough, right?) and the negative to the other outside pin (this was a bit of a guess). I left the middle pin disconnected.
Wowza! The phone turned on! This proved the phone worked.
Since the battery wouldn't charge through the phone, I wanted to try to charge it another way. I connected a 5V AC/DC adapter to the battery pins using tape to hold the wires on the battery. I let it charge for about 20 minutes and then installed the battery.
Yippee, it started up. So the phone is good and the battery is good. There must be something wrong with the charging circuit.
Sure enough, I took apart the phone and found the charging port was broken loose.
Anyway, we bought another phone with a broken screen to use as a charger. Then we can just swap the batteries.
I bought some supplies to try to resolder the charging port. It's really small but should be possible. Look at soldering tutorials on www.sparkfun.com