In addition to Claudia's list of requirements, I like an assurance frommthe owner that the rental is compliant with the local council regulations on short-term rentals.
Many owners list their apartments on multiple sites---Air BandB, VRBO/Homeaway, Holiday Lettings, etc. You can search multiple sites at once on Tripadvisor under "Vacation Rental" and filter for your requirements. A quich look for 3+ bedrooms brought up this one, among others, in your price range:
If you prefer not to rent through Tripadvisor, and/or seek more reviews,myou can Google the exact name of the apartment and will often find the listing on other sites.
Our favorite london neighborhood is the South Bank across from the Tower of London. In apartment listings this area may be called Tower Bridge, or possible London Bridge. The river has a wide pedestrian path lined with restaurants, bars, and other shops, and wonderful views across the river to Tower Bridge, St. Paul's, and other iconic London sights. The Borough Market is there, along with the Globe Theatre, the Tate Modern, and a boat dock where you can catch a boat up or down-river. We could walk to the Whitehall/Horseguards area or Covent Garden in an hour or less. The RV1 bus passes a block and has an excellent route into the center. Or we could cross Tower Bridge and catch a #15 bus straight to the main bus hub at Aldwych (the Strand). Or take the Tube from Tower Hill Station to the museums south of Hyde Park.
During our three-week stay last May we had numerous guests, and they all raved about the location---particularly the ease of walking traffic -free, and the beautiful views across the river.