Londoner here

there's lots to do - would agree with the above poster who suggested Greenwich, it's a lovely spot with some nice pubs along the riverfront for lunch. You've also got the Royal Observatory at the top of Greenwich Park which is a great experience.
I have a few friends that have gone to
https://jackandbeyond.com/pages/bottomless-cake-event which is in Battersea if you were after some good cakes. Would also recommend Spitalfields market for some yummy not too expensive food. Borough Market is also great but is nearly always heaving, and you will be charged £10 for a pot of chocolate covered strawberries
If you're into nature, Kew Gardens is gorgeous (can be quite far out depending on where you are staying) especially if the weather is nice. Same as the Kyoto Gardens in Holland Park, lovely to stroll through on a nice day.
The Tate Modern quite often has exhibitions on that may take your fancy (I know currently there is a Yoko Ono exhib on until September)
The Sky Garden (walkie talkie building, Fenchurch St) is free to get into if you fancy a nice view to take pictures from, they also have a bar for drinks (albeit pricey!).
Lastly you've obviously got your good old museums, Natural History, V&A, National Portrait Gallery, Science Museum (if you have kids) are all free entry

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