Pictures!
Phew, so that took longer than expected.. Two hours including a trip to the hardware store!
I got through the whole procedure, and now I have installed my own home made spring!
Pictures and comments follow:

After following
this guide, I got to the innards of the stick. I didn't know it at the time, but the highlighted thing contains the broken spring. I thought it was the "hat", and that the cap was further down. Confusing!

Another angle. I don't remember how I got the second part of the stick off, but I was careful. I'm not sure I had to though?

THIS is not the thing I was looking for! It looks kinda like bullhorns, so I really though nothing was wrong after all, and started to reassemble... But alas, I the stick was still broken.

I carefully pulled / twerked the whole HAT thing off, until I got it!

I could see part of the spring sticking out. I thought I had to put it back in. Nope. More stuff should be sticking out.

I really struggled with this part, I couldn't see any way to get in there, and the grease was really obscuring what was there. Finally, I saw the hooks and grew an extra pair of hands to open it up. (Actually, I jammed a tweezer in between the parts and pushed the hook with a screwdriver)

The two pieces!

The broken spring!

My first (failed) attempt at a new spring, the old spring and the working spring I made, including new grease! (I think that's important?)

Aaaaand DONE! Only one bit left over. Still not sure if it was part of the joystick originally...
I hope this helps someone later. I have no idea how to turn images around, so... tilt your screen or something.