London hotels are expensive. In fact, London is expensive. If you are staying for a week, AirB&B is by far the best option, but think hard about the location.
Most of the main tourist sights/sites are strung out along the Thames - Houses of Parliament and Westminster Abbey, London Eye, Buckingham Palace, Churchill War Rooms, Tower Bridge, Tower of London, St Pauls etc etc.
There is the West End, (Uptown un USAian), and the East End, (Downtown). The Tower is downtown and even further down is Greenwich with the Cutty Sark and the Royal Observatory etc. The West End has most places, including the theatres and several markets.
Travel around London is easy and not too expensive if you pay with a contactless card or buy an Oyster card. Each member of the party will need their own. Staying in the suburbs will be cheaper, but you will spend more time sitting on a train in a tunnel, so that has to be balanced against the extra cost of a more central location. You may find that places in The East End are cheaper.
Eating out is easy with just about every cuisine you can imagine. When we stay in London, we tend to have breakfast, get some sightseeing done, and then stop for lunch. This means that we only need a light meal in the evening when restaurants tend to be more expensive and crowded. Ready meals, especially those from the more upmarket supermarkets like Waitrose and Marks & Spencer can be surprisingly good.