Here are some picks that haven’t been mentioned.
Music technology > Wunjo Keys has a basement with lots of synths and the guys are happy to set any up for you to play, no eurorack but good to test out newer stuff like Digitone etc.
London Modular > great guys, but ring in advance.
Vinyl / CDs >Phonica records for dance, Sister Ray for everything else ( both Soho)
Gosh Comics / Orbital Comics > independent comic stores with comic art exhibitions and loads of other cool indie stuff. ( Soho/Leicester square)
Books > Foyles. great independent book shop on many floors… like visiting a life before Amazon.
Theatre > Soho theatre does short runs of comedians and other interesting shows. Curzon is a central Arthouse cinema, with a nice bar.
Institute of Contemporary Arts > near buckingham palace if you want to do the whole touristy thing, but great exhibitions/ great cinema/great bar restaurant.
These are all very central so you dont have to make a special visit. other places like Cafe OTO are really great but destinations outside central ( & not necessarily on a underground line)
a day in Hackney Wick/Dalston/Stoke Newington would be great for a trip out to north east
a day in Central London would be great with the shops I have mentioned.
a day south London ending on the Southbank (/ Tate Modern BFI/National Theatre/Old/young VIC would also be really nice)