So I know this is a big ask, but in a perfect world I'd be looking for a small apartment/hotel close (5-7 minutes) to a tube stop and with laundry facilities either in the apartment or in the building. Has anyone stayed in such a place? Does it even exist? After hours on booking.com and other sites, I'm coming up empty.
The Fraser Suites Queens Gate in Kensington has a free guest laundry with a couple of machines. Less than a five minute walk to the Gloucester Road underground station.
Locke at Broken Wharf answers the query completely.
I love Locke at Broken Wharf.
Locke has a number of locations in London, and along with Cove and SACO are aparthotel/serviced apartments brands owned by Edyn. Cove's Cannon St location is next to Mansion House tube station.
https://www.lockeliving.com/en
https://www.staycove.com/en
https://www.sacoapartments.com/
I have stayed in a beautiful flat in Pimlico that matches your description. Only problem is that it books up about 1 year ahead.
Have you looked at VRBO?
Staybridge Suites
Hyatt House
Residence Inn
Citadines
Wilde by Staycity
Frasier Suites
Each has multiple locations in London. They all have in hotel laundry facilities. Some are free, some you have to pay.
Clearly I was looking in all the wrong places. Many thanks, everyone.
OP - another aparthotel brand that may be of interest is Native at https://nativeplaces.com/
My parents have stayed in a flat rented by Studios2Let in Bloomsbury (https://www.studios2let.com/). They have flats in several locations across London, and have access to laundry facilities.