After a little bit of research it turned out that fixing the battery is going to be an easy job. The original package hides four, 1.2V 500 mAh cells, resulting a total voltage of 4.8V. I ran to the nearest shop and bought the cheapest 4-pack of AAA rechargeable batteries. 1.2V, 700 mAh each. Better than the original one!
I only had to solder them serial and the output of the pack to the PCB. After one night the following message greeted me:
Since the solution was working, I electrical taped the hell out of my creation and finally closed the chassis. It does not look good, but it works, it can not be seen and it was cheap!