In your "hood" or relatively close by:
The Leighton House Museum
The Design Museum
Walk through Kensington Gardens past the Round Pond and The Palace, eyes and ears open for the green parrots. Head towards Bayswater Road.
After visiting either of the museums above walk through Holland Park and treat yourself to chocolates at Melt which is near the Castle Pub.
I stayed in an AirBnb property in that neighborhood a few years ago. It was when the Holland Park station was closed for elevator refurbishment so I walked everywhere. Loved the 4 blocks along Holland Park Ave. Neighborhood feel. Daunts Books, Paul's, Tescos. Even the evil empire of Starbucks. You can easily walk from there to Portobello Road if so inclined. Oh should mention in Holland Park is Kyoto Garden complete with peacock.
If an early riser walk over to the Serpentine in Hyde Park to watch the morning swimmers in the Lido and then enjoy your morning coffee outside the Lido Cafe. At 10am the Princess Diana Memorial Fountain opens.
Different from what you've seen in the past is the area around St Pancras.
If you've not seen the Treasure Room at the British Library do so then walk over to the Radisson at St Pancras and enjoy a spot of tea or a cocktail at the gorgeous Booking Office restaurant. Not cheap but dang it's a nice place to sit alone or with family.
Then meander behind St Pancras. Walk over the Regent canal overpass into Granary Square and then into the Coal Drops Yard. This area was the dregs in the 80's and 90's unless you were into raves. Huge redevelopment project. Once you've wandered around there follow the Regents Canal footpath to Camden.
This past January I discovered a tiny bookstore called Hurlingham Books. I'd thought about going to a movie in Putney Bridge but once I got off the tube I spied the teeny Dr. Expresso Caffetteria across the road. So I enjoyed a flat black sitting outside in the cold air and then walked up the road to find Hurlingham Books. Ray, the proprietor was so kind. We chatted for a good half hour before I gave up the idea of seeing a movie. Went over to and looked inside of the All Saints church then strolled along the Thames Path to The Dove pub in Hammersmith. Think it's about a 2 mile walk. Great sage and pumpkin soup at The Dove.
On another trip I walked from Tower Bridge to Rotherhite. Was going to the Brunel Museum which sadly wasn't open. Good walk though. Discovered the Angel and Mayflower Pubs, the Sands Film Studio, Canada Water Studios, and the Surrey Quay. Interesting neighborhood.
Websites you should check before you leave to see if there might be something of interest:
Wiltons Music Hall
V & A Museum
Fashion and Textile Museum
The Royal Court Theatre
The Bridge Theatre
Timeout Magazine London
Lastly if you are at Granary Square there is a Dishoom which had just opened when I went to it.
Probably over run now but Dishoom is a chain so look about for others.
In the Duke of York Square by the Sloane Square Underground Station is Comptoir Libinais. Fantastic spot for lunch. Believe its a chain as well.
Another fun thing to do is to check church bulletin boards. Might find a car "boot" (trunk) sale, book sale, recital etc. I've mentioned it before on the Forum but one year I was at the Royal Academy of Art and overheard a woman mentioning the artists open house on Eel Pie Island. Who can pass that up? Took a train from Waterloo. As I wandered I was surprised to find the York House Gardens. Brought a smile to my face as I had no clue they were there.
Don't forgot www.walks.com either. Can never go wrong with a London Walk!
Upload the CityMapper app on your phone, grab your Oyster card, wear comfortable shoes and off you go.