Our first few days will be in Kent. That (and the tomb of both a prince and a martyr) piqued my interest in Cantebury. Just saw the pictures of Salisbury. It is jaw-dropping! As that's close to Avebury (with the better stone circles that you can actually walk around, reportedly,) I'll be sure to make an effort to take that in.
My kids will 4 and 6 at the time of the trip so I plan on half-day trips and liberal use of the tablets. Good to know about the free museums as my son is 90% dinosaur in his head and what castles are worth it.
Where in Kent are you staying?
There is some really crap advice in this thread, Rochester..... no mention of the Cathedral. It's not quite Canterbury but it's there and right next to the castle and the Chatham Dockyard is very close by if you wanted to make a day of it. Hastings..... why? Bournemouth.... why? Neither of them are in Kent. WTF is the big yellow ring around London that's to be avoided too? Dover is a town and not a city. I'd be tempted to ignore
treelo and
treelo because they're full of shit.
Stonehenge is indeed crap but it's a 2 hour drive from London but if you want to go to Salisbury combine the two or go to Bath or Longleat Safari Park but you'd need a car and getting in and out of central London at the sort of time you'd be travelling for a day trip is probably even worse than you think it is. Salisbury is very possible to do in a day trip by train though but best not to bring any novichok with you and to actually visit the cathedral.
You mentioned Hadrians Wall in your first post, that's a 6 hour drive from London.
Most of it's been covered by the more sensible posters here, the Kensington museums (Natural History, Science, V&A), Tower of London and Tower Bridge, Buckingham Palace, the numerous parks, Greenwich (park, observatory, maritime museum), from Greenwich get the cable car across the Thames and probably come straight back to Greenwich on it or get the DLR back into London as there's nothing on the North side of it, Windsor (Castle mainly but Legoland is there too), House of Parliament (but Big Ben is covered in scaffolding), Kew Gardens, The Sky Garden is free but you need to book it and not bad for a decent view of London, the View from The Shard, The London Eye, HMS Belfast, St Pauls Cathedral (pay and go to the top), British Museum, Museum of London, The Great Fire Monument (it looks a bit like Nelsons Column but you can climb to the top), Trafalgar Square, The Tate, The Tate Modern, 221b Baker Street is worth a look for, London Zoo, there are a couple of urban farms which would probably work for a 4 and 6 year old so Mudchute or Surrey Docks Farms......
Or you could do what most tourists with young kids seem to do, spend all day on the tube getting in my way.......