Hello everyone!
Made an account just so I can express my gratitude and say a big THANK YOU to everyone in this thread who taught me how to replace the battery and where to buy a new battery from.
Just in case anyone needs this info, I will shortly take you through our experience. I asked my brother to help (he pretty much did all the delicate work and he's my hero).
1. We bought a 3,7V 50 mAh 360821 Li Polymer Akku Battery for 8£ from ebay
2. Just like you taught us, we began by removing the black cap and unscrewing the bottom silver part of the Sonic Screwdriver (righty-tighty, lefty-loosey, so we went counter clockwise)
3. The unscrewing proved to be difficult in our case as there definitely was a bit of glue holding it in place. Going back and forth firmly but not forcefully finally made the glue give in and we managed to unscrew the silver part.
4. The white casing inside was held by a small plastic locking mechanism. We used a small blunt metal object to push in that "square" and release the grip.
5. Gently splitting the white casing open, we felt like it was opposing quite strongly. We couldn't open it as much as it was shown in the pictures you guys posted. It felt like it was gonna break if we tried, so we kept it open as much as we deemed "safe".
6. Using various small metal tools, we tried getting the old battery out. It was heavily glued to a small plaque inside. So we carefully kept pulling and wiggling it. This took a long time, especially as we tried to not pierce it with our tools. Finally, we managed to remove it, squeezing it (barely) through the white casing. The plaque too. We were careful not to damage the wires or pull too hard.
7. We unwrapped the yellow ducktape and exposed the location where the wires connect to the old battery. Using the soldering gun, we disconnected the wires from the old battery.
8. We did not have enough space to solder the new battery onto the circuit board, so... using a sharp knife, we cut the wrapping/cover of the wires, exposing more of the wires, on both the new battery wires and the wires from the old battery which were still attached to the circuit board.
9. We twirled the wires together (red with red, black with black) and used the soldering gun and soldering metal to fix them together (red to red, black to black). We used the charging cable on the sonic and it lit up! Red light, no more "instant green because old battery is dead"!
10. We removed the charging cable, gently shoved in the new battery, closed the white casing, screwed in the silver piece and connected the charger again. It still worked! Two hours later, sonic is functioning good as new!
Very happy (especially for someone less handy with tools as I am). Thank you again and hope this helps anyone in the future
Cheers!