I'm 8km from the school I teach at. That's just enough to make me sweaty. Showering at school is perhaps possible (there is a tiny faculty locker room on the other end of campus) but not really realistic.

I'm considering an e-bike to make the idea more realistic. A couple more miles per hour with a little less effort changes everything.

A bigger problem, unfortunately, is that I'd need to travel along a really busy highway at each end of the journey, and it kills cyclists with some regularity.