The difficult and dangerous operation is the tag connection to the Li-ion battery case. The Weller soldering iron shown in (7) has the wrong shape solder tip. The connection to a Li-ion must be made quickly and without significant heating of the Li-ion. Connections are usually spot welded but an amateur could try a solder gun.
The replacement batteries should be Li-ion size 18650 of at least 1500 mAh and identical. These are available with tags already attached and I would buy these and crimp to the tags. I have not done this myself because I found the receipt and the drill was bought 3 years ago so it is going back. The batteries were manufactured 4 years before I bought the drill so they were old when I bought it.
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