Replacing an IKEA FIXA cordless Screwdriver Battery

Got one of these? For 9 € with your latest IKEA furniture purchase? It did a good job until lately but now won't charge and turn? Time to replace the battery, if you can get one for cheap, otherwise it's not really economically sound to do so.
I did it anyway.

Note: The repair has been holding up for a year of frequent use, but a few things came up:
Reuse the paper housing for the battery cell. It really helps and protects the battery inside the fixa. No need for glue or anything then!
It might happen that the button gets sticky and the Fixa wants to stay on. This is due to the high currents combined with a cheap switch inside. You can open that switch carefully with a thin screw driver and clean the contacts from burn residue and it should work again. Everything only fits one way inside the switch so you can't really mess it up.
The power supply to charge the Fixa has died. It's probably due to the long charge time with a cell twice the capacity (original charge time was already many hours, there is no quick charge). Luckily I did have a spare.