Since they do ask for a “walkable to major sites” I will heartily endorse Claudia’s suggestion of the Westminster/Whitehall area, and specifically the Royal Horseguards Hotel. I was going to suggest historic St. Ermin’s, near Westminster Abbey, which we enjoyed very much, but I have to admit I would prefer the location of the Royal Horseguards for one major reason which I will explain later. I will still post a link to St. Ermin’s so you can compare. It has a lot of history—-but so does the Royal Horseguards Hotel. August prices for both appear to be well within your budget.
https://www.sterminshotel.co.uk/
https://www.guoman.com/the-royal-horseguards
But first, to address the “walkable” idea. Yes, London is large, and the main attractions are quite scattered. However, for the usual suspects”, namely the Tower of London, St. Paul’s cathedral. Trafalgar Square, Covent Garden and the West End theatres, the British Museum, Churchill War Rooms, Westminster Abbey, Hyde and St.,James Parks, Buckingham Palace, the Kensington museums (Victoria and Albert, Science Museum, and Natural History Museum), and the South Bank sights such as the Tate Modern, Shakespeare’s Globe Theatre, and the National Theatre complex, this area of Westminster (or Whitehall) is actually “central”.
And it is not only possible but enjoyable to walk to each one of the places listed from a hotel in the location of the two suggested above. I know this because we have done it.
Yes, some of those walks will be 2.2- 2.5 miles. That is one hour of healthy outdoor activity. Something we much prefer, over riding underground in a crowded tube. Most of our London visits we avoid the Tube altogether.
For sights to the west of St. Ermin’s, we have a very nice route through the parks—St. James, with its lake and beautiful plantings, then Green Park, all lawn and trees, then Hyde Park, with the rose garden, large lake, and Princess Diana memorial fountain. For sights to the east, we cross the Thames** and use the very pleasant pedestrian Thames Path, with its great views across to classic London buildings. If the destination is on the north side of the river, such as St. Paul’s, we use the pedestrian-only Millennium Bridge to cross back, or, for the Tower of London, the iconic Tower Bridge which has a good pedestrian path well-separated from the traffic.
If for some reason we get far away on our walk and do not have time or energy to return on foot, we use the London double-decker buses, or a Thames commuter boat ( now called Uber boats) to return home, disembarking at Westminster Pier.
Here is a good schematic map of the London bus routes that connect the major attractions:
https://content.tfl.gov.uk/bus-route-maps/key-bus-routes-in-central-london.pdf
On our numerous trips to London over the past 20 years, we have walked as far as Greenwich to the east and Chelsea to the west, returning by boat in each case. London and Venice are our favorite European walking cities. I recommend you look closely at these or other Westminster hotels—-by which I mean not all of Westminster but specifically close to the river, and the Elizabeth Tower (aka Big Ben) or Whitehall.
** And now why I like the location of the Royal Horseguards Hotel even more than St. Ermin’s: we detest crossing the Westminster Bridge. It is way too crowded, and often blocked with throngs of people watching the scummy shell games. The bridge we like it the next one downriver. The Hungerford Bridge is a railway bridge, but it has separate pedestrian bridges, the Golden Jubilee bridges, on either side, making for a pleasant and scenic river crossing. One can even stop and take photos.
The Royal Horseguards hotel is right at the base of this bridge. Of course, one can access the pedestrian bridge from other hotels in Whitehall or Westminster, but this one makes it very convenient.
