In Order to remove the battery which is of course held in place by adhesive you should use high amounts of rubbing alcohol.

Give it a good squirt, wait a little and then slowly try prying the battery out. Do not bend it, especially when fully charged. Pry - use more rubbing alcohol - pry - alcohol ... and repeat until it comes out unscathed.

The flat ribbon cables run underneath the battery. With all the alcohol used you should be able to tear off the right one with ease.