I figured it out. I guess you have to use the port 8081 instead of 8080.
So if anyone reads this, the address for the kodi web interface is: [ip-address]:8081
Is this really common knowledge? Otherwise I think it should be stated somewhere on the website or in the manager.
I found it out by searching the forum a bit more thoroughly and came up with a comment mentioning a now non-existing github link.