Use your soldering iron to heat one of the two battery terminals on the back-side of the top circuit board. The terminals are conveniently marked.

When the solder is melted use your plastic spudger to pry the battery away from the top of the circuit board.

Repeat for the other battery terminal and remove the battery for recycling.

Caution! If you want pull the battery away with pliers, wrap the pliers with electrical tape to avoid shorting out the battery. You can probably use your fingers (if not too hot) or your plastic spudger.