This is another idea for a day out from London, slightly unusual, which I have just been reading about.
Sevenoaks has a frequent train service from London (20 minute journey, or thereabouts), and it is then a 20 to 30 minute walk through the town. So very easy to reach
It was built in c 1450 for Thomas Bourchier- Archbishop of Canterbury, then passed to King Henry VIII in 1537 and then to the Sackville family who still live there, but since 1947 it has belonged to the National Trust. From the longer article I am reading it looks very interesting.
It was also the setting for the filming in January 1967 of the Beatles' videos that accompanied the release of "Penny Lane" and "Strawberry Fields Forever".