Do you have a picture of the battery for your screwdriver? I'm not familiar with that model.
It almost certain has a standard size cell inside the battery pack that could be replaced.
The leads might actually be spot welded on, and won't come off as easily as soldered leads, and my understanding is soldering is less than ideal because it takes a lot of power to get the solder joint made before you start to get the internals of the battery too hot.