Metropole would be a good option, given the two tube stations nearby. All the more so if you take buses for some of these trips since the bus service is every 2-3 minutes to many of these places. You just need a contactless payment card each. Other possible options being Paddington, Park Lane, Waldorf, Bankside, Tower Bridge. Waldorf isn't that near a tube station but it is central to begin with, Bankside and Tower the same, but further from the main sites. Conrad also fits your criteria and is both central and on top of tube station. There are many other Hilton branded properties by tube stations, but typically they don't have a lounge (Trafalgar) or are further out (Islington). Given what you have said, I would focus on Metropole, Paddington and Park Lane. Metropole does have some "family" rooms in the West Wing, which sound like they would work for you.