I finally got it working.
I ended up doing a factory reset and starting over from scratch. It connected to the router on the first try after the setup was finished. The auto detection worked, I didn't have to manually create the connection.
I was rather annoyed to discover that you cannot set a static IP address. Due to the way I have my network set up, I can't use DHCP on my N routers (they're set up as access points), so I had to set up a DHCP server just for the Touch.