A new shower and locker facility,
Run Pit by au Smart Sports, has just opened next to the Imperial Palace.
You just bring your exercise clothes and shoes. They have lockers and showers. It works like a gym with no machines.
Fee: 800 yen / day
Hours: 7-22 (Weekdays), 8-18 (Sat. Sun. National Holidays)
location:
Palace Side Bldg. adjacent to Takebashi subway station.
Sample jog on YouTube
Looks like there are some other similar facilities nearby.