Like most Li-Ion battery packs, the cells are connected via spot welded metal strips. To remove the cells from this pack, I unsoldered the sensing wires, and then used a wire cutter to cut the strips as they wouldn't be reused.
The strips are spot welded to the battery terminals because it is not only difficult to solder to them, it's really not a good idea. There is a good deal of metal to heat up, and the cell itself is not meant to take that much heat and may be damaged. This is one of the advantages of the cells that I sourced, they already have wires on each end.
More specifically, they have a small tab welded on(remember the note about not soldering to the battery itself?), with the wire soldered to the tab. These would be much easier to assemble.
I inserted the cells into the frame in the same orientation as before:
It was very easy to solder the wires between the tabs to make the 5S string.:
I probably should have replaced these wires with thicker gauge, and would do that next time for the next one I rebuild.
I used some tape to hold the frame together, taped the temperature sensor back into place, and reinstalled the controller board. Then it was a simple matter to reattach the leads and voltage sense wires. I'm not sure if the battery monitor needed a specific power sequence, but to avoid damage, I started with ground, and then did the sense lines in order before attaching the positive.
I put the foam back into place, and fit it tightly into the case. Prior to starting the project, I fully charged each cell in it's original USB charger so that they would be roughly balanced. This was a cell voltage of between 4.1-4.2 depending on the charger. The variation was likely due to tolerance on the different charge chips. The on board battery level meter in the pack showed full right away.
Testing out in the drill, it works great, and charges correctly on the drill's original battery charger.
Overall, this project cost $10 plus an hour of work to rebuild an $80 battery back to it's original performance.