Where is it? 5 Strand Lane, off Strand/Surrey St, hidden away in a back lane and probably half a building site.
What's quirky about it? It's a National Trust property which is the "remains of a bath - possibly Roman". To see it, you press a light switch on the outside wall and peer in through a grimy window with a grille. You'll see something like the picture on the right (nicked from Wikipedia), or possibly just a grimy window and a grille. Apparently you can make an appointment with Westminster Council on 020 7641 5264 to view it, free, though don't say you're going by bike: they're a bit anti that sort of thing.
Why bike there? There's lots to see and do in Strand and Fleet St, and Somerset House is almost next door.