Yeah I was going to say I see plenty of listings on VRBO around Hyde Park advertising washer/dryer but these are small apts. so they're probably combo units.

In France and Italy, you'd find these places which would wash, dry and fold by the kilo so a week's worth of laundry would cost 10-15 Euro but in expensive London, who knows how much it would cost.

I guess I didn't explore Hyde Park enough. Didn't really grab me that much but I love the area around Royal Albert Hall and of course the V&A.

Oxbridge Apartments in Kensington and Chelsea fit your price requirements and include combo washer and dryer. There is also a launderette that also does service washes practically across the street along with a Sainsbury's super market. Tube station is not far and Hyde Park is walkable.