I did it! I was so nervous until I had put it back together and tried it and saw it worked... I got to use the soldering station of a friend instead of having to buy a cheap one.
Since I hadn't soldered before, the already defective *istDL proved worthwhile to practice on while taking out the two good white solenoids. They were pretty hard to get out since there was something else but solder around the cables, I don't know if it was hardened glue or some kind of plastic, but it took a while to get that hot enough. In the process I managed to touch the white plastic body cap which I only recognized when a long string of soft plastic got moved along with the soldering iron I wanted to put back
It's good that it happened then, so I was much more careful with the K-S2. Getting the green one out of it was easier than with the *istDL, and I'm not sure if that was because of the glue or what it was on the *istDL that was not present on the solenoid of the K-S2, or because I used a little higher temperature than when I was practicing on the *istDL.
Getting the old solenoid out was not as hard as getting the new one in, it must have fallen down three or four times before I got it in place.
Holding the cables in place in order to solder them on proved to be a challenge because of the tiny wires and space constraints by the body, but using a "third hand" clamp I managed to stabilize them good enough, even if the pink cable was squished hard by the clamp.
Soldering them on was the easiest part of the whole process I think.
Here's a pic of the new one in place after the fact:
The one on the left is far from perfect, but it's holding on good enough after trying to pull on the cable. I think the one on the right looks quite good?
After putting everything back together (the little washer on the microphone port did fall off at one time, but I saw where it came from), putting a card, battery and lens on, everything worked as it should again!
I want to thank you @photogem once again for those excellent, detailed instructions you've decided to share with us! I wouldn't have had the confidence to try this myself if it wasn't for this very thread!
Oh and something else which surprised me a bit: I hadn't realised just how big that flash condensator was... that's a lot of space to put some other tech in when you decide to produce a camera without internal flash...