I found a way to do it without really needing any tools, just a screwdriver and long-nosed pliers basically. It's hard to explain, especially in my not-native language, but I'll give it a try:
With the first of my two pedals, I made it the way it's described by this video by Ernie Ball, which wants us to put in the back string, put the spring on it, and then use the hook to tighten it. Then the same with the front string, put it in and then do it around the poti 3 times, then put the other end of the spring to the string and after that tighten it all up by putting the string around that roller thing next to the poti.
What I did was, I removed the mount of the poti and made sure, that the strings and spring stayed connected. If you then watch this, you can see that you could connect those, BEFORE you put the front string around the poti. You only have to make sure to turn the front string 3 times to the left. What you do then is to make sure the spring stays connected to the strings, while you put the front string around the poti as nicely as you can, already going around that roller thing. Then you make sure you can put the mount in place to be screwed down to the case. Fine adjustments can still be done at this point by pulling on the spring while turning the pedal.
Oh man this is difficult to explain ... Maybe you get the idea, that's all this post was about. Using this method it only took me 5 minutes to reassemble my pedal and it's perfect!