Golden Hinde, check. Cutty Sark, check. Mayflower? Did I miss something?
You don’t have to go inside to stand on the line. It goes all the way round!
I agree with everyone who’s mentioned the free museums. There’s no earthly reason to spend any money on attractions in London. Natural History Museum, Victoria and Albert, Science Museum, British Museum, National Gallery, there’s your weekend full. Especially with some strolling in parks thrown in. I also agree that you don’t want to actually stay over towards Kensington. I like South London, but then I would, living here. Either way, Kensington’s soulless and expensive.
Transport is very good everywhere. I’d avoid the Tube unless you’re in a real hurry - take the bus, get a feel for the place. See stuff. Skip the tourist buses - with an iota of sense and general knowledge, you’ll know what you’re looking at, you don’t need to pay six times the price to have some bored teenager telling you to look left for the Tower. There are several bus routes taking in a good range of sights (one example below) which you could find out about in advance and impress your family with your encyclopaedic knowledge of London as you guide your own tour! You also, by the way, don’t really need to see inside the Tower. You can walk round the moat and go “hey, it’s a thousand years old!” and then go and spend your money on something else. Unless you’re a major fan of the Baroque, don’t pay to go into St Paul’s either.
The Horniman is good; it’s not free, but nor is it expensive. If you wanted a slightly more relaxed day involving not being on your feet around museums for all the hours, you could see the Horniman in the morning and then have a ten-minute bus ride up to the site of the Crystal Palace, wander the terraced park with views across London one way and the hills of Surrey the other, see the 1956 transmitter tower which replaced Logie Baird’s original station, see the Victorian dinosaurs and Pissarro’s house, and get some decent food a lot cheaper than in the centre of town - or have a picnic in the park. Then take a different bus back to the centre through Brixton (famously dodgy home to David Bowie, Van Gogh and Havelock Ellis - I just randomly selected 3!), past (or, you know, go into for free) the Imperial War Museum, past Lambeth Palace where the Archbishop of Canterbury lives (and which is much, much prettier - and older (1490ish) - than Buckingham Palace (not even finished until the 20th century), which is an eyesore), Westminster, Trafalgar Square (where you could hop off for the National Gallery), Piccadilly Circus, up Regent Street and finish at Oxford Circus.