Neither 'åke' nor 'kjöre' would work in Norwegian. You should always use 'kjøre' in that context. (or 'kjoere' if ø is unavailable, but neither ö nor oe works as replacements in Duolingo afaik)
I've never heard anyone ever use the word 'åke' in Norway, so I wouldn't recommend you using that either. It's listed as a loanword from Swedish in my dictionary, but not with the meaning of 'drive'.
I do use the correct characters with exercises, since the special characters are displayed (and clickable) below the exercise, but obviously that is not the case with comments ;) My dictionary (from 1996) claims 'åke' to be driving, and then has a phrase 'åke inn' (to be sent to prison), so I'll just throw Åke away.