Thanks for your reply, but googling with the exact search term you note was the first thing I did.
The closest one is a 14 minute walk from our hotel... not on a train line, so that's 14 minutes walking with laundry.
Not ruling it out, that's what rolling suitcases or taxis are for.
Was hoping someone in the area knew of something closer, that is not on the map.
We're staying at the St Regis.
I posted on the "living in Japan" forum, because locals living in the area are more likely to know something like this!
Thanks anyway, for the help. I'm sure your post will help someone else in the future anyway!