And here is my recommendation for a place where you can grab a bite. Georges is a fish and chips shop loved even by Jamie Oliver and many celebrities, you can see some of their photographs hanging on the wall when you first step in from the door.
Another location worth hanging around is beautiful Chelsea, in particular Sloane Square and pathways and parks next to the river. If you decide to take a stroll down Kings Road, coming from Sloane Square station, make sure to stop by Joe and The Juice. Super food smoothies and shakes with the opportunity to sit outside and enjoy the sun. Going for a bike ride is always an amazing idea! Also as currently The Chelsea Flower Show takes place, almost every shop has a cute flower window display and the whole road is suddenly transformed into this fairy like place.
On a more serious note, If you fancy getting to know London from another more different time, I suggest You get yourself "Night walks"
But be aware it is in the style of the author, so nothing pinky and romantic. It is rather realistically dramatic.
Dickens describes his time as an insomniac, when he decided to cure himself by walking through London in the small hours, and discovered homelessness and vice on the streets.
I feel it is a very interesting contrast of what London can offer.
Shop here: Night Walks (Penguin Great Ideas)