I don't live there, but my solution in a similar situation revolved around the exercise club two blocks from work. On bike days you only use the club for its shower, on non-bike days (winter, etc) you exercise at the club (lift weights, mostly).

Also, hard as it might be to believe, bicycling more energy efficient than walking at the same speed, so once you're in shape, going slow and enjoying the ride is not sweaty, its possible to go slightly faster than walking while generating less heat than a normal walk. Depending on your local weather, a leisurely walk outdoors often results in no sweating at all.