I think three hours is simply too short to make it worth it. If the main purpose is to take a shower and one does not have access to showers at the airport or train station, one could simply buy a day pass for a fitness center / spa / pool, get a workout in and shower there.

If the room was available for longer and would allow for a decent amount of down time, I could see the case during a long overlay at an airport. AFAIK, some hotels near airports offer day rooms for a longer period of time.