The green goes to the firing pin aka center pin and the little black goes to the outer shell as ground. Honestly it is not worth saving this battery as for they are dangerous especially once a user takes them apart. I am the son of an electrical engineer and my sister is a EE student and I am a mechanical engineering student. I have confidence in my knowledge on soldering and electricity to do the work but I CANNOT suggest anyone else to do this without proper knowhow. Soldering directly onto the battery is dangerous, very dangerous. Do you just have a cheap soldering iron or do you have a temperature controlled station? If you have a cheap iron that just plugs into the mains/wall outlet then the temperature can't be controlled and damage the battery which can basically turns your
device into a time bomb with no one knowing when it will blow. The medical bills will cost more than buying a new
device. If you decide to head my warnings and try and fix it. Try and find a fine tip soldering iron. DO NOT apply the soldering iron for more than 2 seconds at a time to the battery. When soldering to the board be careful, adding too much solder and causing a short. Plug in your iron and let it heat up. while it is heating keep testing the temperature with a piece of solder. When the solder melts then solder to the battery that way the ion is just hot enough to melt the solder and not too hot.
I do not recommend fixing this
device if you do you are doing it at your own risk.