Hi - I lived most of the first 40 years of my life in London, then quit London in 1978 to live in Devon, in the South West of England.
What is 'cool' to one person, may be boring or disgusting to another person, so it is rather hard to suggest places without knowing a bit about what might interest you.
When I was in London, our family lived in Hampstead, in the northwest of London. Hampstead was traditionally an area for writers, poets and artists: George Orwell; John Keats; John Constable, and many others. It is also famous for its Heath, a large area of natural parkland, woodland, many ponds, and Parliament Hill. This is a high hill that has views across London, and one can see St Paul's Cathedral, the Houses of Parliament, and modern things like the London Eye, and the new Shard building, which, of course, did not exist when I lived there.
During the 1800s Hampstead Heath was threatened with housing development by rich landowners, and there were many arguments until it finally became a protected area for the benefit of everyone. We moved there in 1945, at the end of the War. During the War much of the Heath had been used to grow crops, and graze sheep, and I can member, as a child, seeing those things. A bomb had hit a children's paddling pool, and it was many years before it was rebuilt. An important railway line, vital to war transport of munitions and troops, was at the edge of the Heath, and German bombers were aiming for that when the paddling pool was hit.
Near Hampstead is Highgate, another hilly area, which is famous for being where Dick Whittington, the first Lord Mayor of London, walked on his way to London. (Hundreds of years ago, Hampstead, Highgate and many other places which are now part of London, would have been villages in the countryside, a few miles from London). Beyond Highgate is Alexandra Palace, on yet another hill, which in the 1930s was the home of BBC TV, probably the world's first public TV service I believe.
I will give you some 'links' to some of these places. There is also a UK company that sells large-scale maps of 100 years to 150 years ago, which are very interesting to compare what areas were like in those days.
en.wikipedia.org/wiki/Hampstead
www.hampsteadheath.net/
en.wikipedia.org/wiki/Dick_Whittington_and_His_Cat
en.wikipedia.org/wiki/Alexandra_Palace
www.alangodfreymaps.co.uk/london.htm
I hope these links work for you; if not, you can of course Google for yourself all the places I mentioned. The Internet gives one the ability to find out about, and even 'visit' via Google Street View, places you are interested in which would have been totally impossible a few years ago.
Good luck with your holiday.