I've got it to work! Finally and THANK YOU!
Here's what I did:
Ok, I looked for that Intel Connection Software and found both it and a new driver for my card on its site.
After installation:
When I tried to activate the IC Software it said I needed to do it manually first. That meant hitting Fn+F1 (took a while to figure that out).
I also stopped and inactivated the Wireless Zero Configuration (because it's run by Windows) found in Control panel -> Admin tools -> Service (or whatever the correct English words are, I'm Swedish).
I activated the ICS again, which worked for about a second. I tried to repair the wireless connection, which worked. The Intel(R) PROSet then found the network and I got connected.
Happy, happy me.