Remove the kapton tape securing the new battery's circuit board

Use a soldering iron to remove the attached wires from the circuit board's pads

Use a soldering iron and some soldering wick to clean up the pads

Add some fresh solder to the pads and solder two thin, flexible wires to the positive and negative tabs of the circuit board

Please take note of the wires' orientation. As you'll have to wrap the wires around the battery to fit the battery and wires into the case, solder them at an angle to make your life easier