(It's a backup battery, actually.)
Now that it has booted, leave it plugged in, shut it down, then plug in your backup battery. If the main battery is dead, leave it out of the machine. Start up again, put to sleep, and keep it plugged into mains power like that for a few days.
Most Pismo backup batteries will revive that way, and will charge whether there is a main battery in or not. I doubt that you need to spend so much a penny on it.