You may have to up your budget (depending on dates), but the St. Ermin's Hotel in London was absolutely perfect for my family of four. It's tucked away on a side street adjacent to Westminster Abbey. We were able to walk to the London Eye, Big Bend, Churchill War Rooms, St. James Park, Buckingham Palace, Trafalgar Square, and the West End theatres. We used a taxi service and river boat to/from the Tower of London, the British Museum, and Hamley's.
The staff was amazing with our kids (aged 10 and 8 at the time). The hotel has a secret agent/spy history and plays into this theme with the kids. Upon check-in, the kids were given a mystery game to play throughout the hotel, and when they cracked the code were given drink coupons for the bar. The bartender gave them a selection of completely over-the-top cocktails and then made a show out of the preparation. Great fun! The hotel also has a good restaurant, top ranked afternoon tea experience, and had complimentary wine on weekday afternoons.
It was a great experience for my family. I highly recommend.