@meeera: I didn't hold the Home-button long enough, that's why the factory reset didn't work. But I managed in the end and the disco lights came on.
Thanks for the tip!
Thankfully, we have a Windows laptop as well as our Linux computers, so plugging the factory reset Kobo in and doing the upgrade via the Desktop App ended up working. This seriously made me realise how much in love I am with my Kobo. A day without it would be torture!
Now I got the latest upgrade and everything works beautifully (even though I miss the little black lines indicating how far I am in a chapter).