I find almost all of London interesting. It's not a city which is dependent on its set-pieces. You can be far from any establishlished tourist sites and still find lots of stimulation. I think it is visually and texturally very rich. The built environment is interesting. The people are interesting. There's a kind of event density that means there's always something happening, alwAys something unfolding, so you can really just walk and be led by instinct and impulse and just immerse yourself in the place. These very distinct, sometimes quite dystopian, horrifying, locales, with quite sharply demarcated boundaries, for instance to come out of Shoreditch with its moneyed, fashion conscious youth and into Ridley Road market with the old Rastas playing dominoes, or from the advertising and media haunts of Soho into the retail inferno of Oxford st. Or the middle eastern Edgware rd running into the grotesque display of Knightsbridge, women stepping out of chauffeur driven Bentleys. Young men driving super cars up and down the road.
We have very lovely outdoor spaces like Hampstead Heath and Greenwich park, Regent's Park, Epping forest and Richmond with their populations of wild deer.
We have the British museum which sometimes feels on the verge of overcrowding, but still is just about manageable and almost without any queue to get in. The museum of London is well worth a trip also.
The National Gallery is full of treasures. You like Rothko and they are in the Tate Modern near where i work. Tate Britain I'm also very fond of and has the William Blake stuff. then there's the natural history museum, the science museum, the Victoria and Albert.... But it's possible to have too much of museums. They're exhausting.
As Tea says there's almost nothing ancient above ground bar a crumbling section of Roman wall and Cleopatras Needle, which is Egyptian. Bits of Norman architecture exist integrated into more modern buildings as in the Tower of London and various churches. It's possible to do day trips to places where you can get a better sense of that sort of thing. E.g. Winchester (we've got a thread about the place) or even Avebury, just about. My inclination would be to stay in London though unless you're feeling claustrophobic.
As with Anglo Saxon London there is very little junglist London left now Blackmarket has closed its doors. I literally can't think of anything. I would consider Hackney the heartland and it does retain something of its spirit but it's also been gentrified to within an inch of its life. The people there are disgusting and hateful. (John Eden being the exception)
Food you have be canny. The golden rule is avoid any resturaunt hyped in the media, traditional or social, n media, newspapers, magazines or Instagram. The hype is always a consequence of PR campaigns, bribes, friends in high places, nepotism and corruption. London suffers from this to an unusual degree. If you can dodge those pitfalls there's plenty of nice things to eat.
I would urge you to climb something to get a view out and over the place. Options include, St. Paul's cathedral and the shard (both expensive) the monument which is cheap, the hills (primrose, parliament, Greenwich) or the new one change shopping mall rooftop, all of which are free. It's quite a spectacular sight. Very powerful. The city of London, what we call the square mile, crackles with occult energy.