Commute times reduce rapidly over the fiorst week or so as you figure out the best route and the best ways through your route. It also becomes safer when you know all the potholes and hazards leaving more brain capacity for the unexected. (always expect the unexpected).
Your bike looks fine for the job, consider adding full-length bolt-on fenders and rear rack. When you become more confident, some toe clips will add some efficiency for very little outlay.