I want to thank everyone for their help and suggestions.
I ended up buying this replacement battery:
I couldn't manage the headset, the battery, the soldering iron, and the solder with just two hands, so I ordered this "third hand" gizmo:
It's not the easiest thing to use, it has no case, and in order to put it back in the box it came in, I have to take it apart again. But for $7.35, it's a good deal.
Once I got it set up, the job was done in about 10 minutes. I didn't even need any additional solder. I was able to melt the little drop that was already there and attach the new wires. I then reassembled the headset and charged it up. As far as I can tell, it's working.
Thanks for the help.