First I'd check with the janitorial staff to make sure there really aren't any shower facilities. I worked in a few office buildings where none of my regular co-workers thought there were any showers but they actually did exist. One shower was quite hidden in the basement behind some of the building HVAC equipment while another was tucked behind a door labeled "Mtce. Staff Only" in one of the rest rooms. The showers were for the janitors who sometimes needed to use rather caustic chemicals in their duties. Once I knew about the showers I encountered no resistance in using them after my morning commute.

And I second the suggestion to use baby wipes or just small towels/soap to do a quick 'sponge bath' in a regular rest room. Fortunately I had enough space to store suit coat and dress shoes in my office so could minimize the clothing that I needed to carry on the daily commute.