My wife and I are planning a London trip for sometime in the Fall. We stayed in a well-located flat near the "border" (if there is one) between Knightsbridge and South Kensington on the (I'm trying to remember without a map) Brompton Road between Harrod's and the V and A Museum. Our flat was conveniently located for the Underground (we usually used the Harrod's entrance, for a couple of green grocers, and for what seemed like a couple dozen restaurants, including the ubiquitous Pizza Express which surprisingly makes an incredibly good salade Nicoise. This flat may not now be affordable for us after a recent renovation, so we'd like to look at a couple of alternatives in the neighborhood. We've had three good experiences with London Connection and may well work with them again.
We hope that some travlers here can suggest some additional choices for us in this location. We are a retired couple and are looking for a flat with a lift and (to save some non-productive work during hours better spent seeing the unlimited sights of London) a washer/dryer in the flat.
I doubt that you will find a flat which is significantly less than the Rembrandt hotel, particularly if you are fortunate enough to be flying British Air and can book through them. I scoured the area for a week long Christmas stay and found nothing to compete with the Rembrandt.
My husband and I stayed at the Crowne Plaza, South Kensington, three years in a row. It is directly across from the Gloucester Tube. We could take the Tube over to the Knightsbridge stop and go to Harrod's to eat. It was a lovely hotel in a picturesque location. This last year, there weren't any rooms available, at the end of June. We stayed at the Doubletree, Westminister. That was a great hotel in a good area also. We could walk to Westminister Abbey, Big Ben, Westminister Pier, the Tower Bridge, Horseguards Parade, Parliament, etc. At either hotel, you earn Priority Club or Hilton points.
Have you tried using vrbo.com? You can designate both neighborhood and elevators in your search.
Through VRBO we ended up in a flat at Nell Gwynn House a few years ago. It was conveniently located in that general area, Sainsbury across the street, bus stop out front.
We've used Home Away to rent a flat in London before.