Having been traveling to London off and on since my college days my appreciation is based on my sense is that it always changing yet always the same. Its architecture, traditions and inhabitants prove that.
To appreciate the dichotomy of London's architecture after enjoying the venerable Tower of London walk up to 20 Fenchurch Street where the Walkie Talkie building is located ( Brits seem to enjoy naming their more modern buildings.) The Sky Garden is atop it. After visiting there stroll up Lime Street up to the glass covered Leadenhall Market which is adjacent to the Lloyd's of London monstrosity. An intriguing look at how old and new meld. (Presently glimpsed in a Jaguar commercial airing here in the States.) You can also see this contrast by stopping at my favorite pub, the lovely Art Deco gem, The Blackfriar. From the street view how it contrasts with the Blackfriar underground station across the road. VERY different. Love the look of both at night.
A great way to immerse yourselves is to visit markets, attend a church service, enjoy theatre, walk in the parks. I love Portobello Road market. I stayed in an apartment and walked 3 blocks to meander the entire length of the market. Easily spent 2 hours ambling, watching, listening, and experiencing this enjoyable market. Arose early and while I drank my coffee and had a pastry at Gail's Artisan bakery watched the market come to life. Its not a daily event so check their website. Engage with the artisans and vendors. Love Brick Lane environs as well. Brixton market has grown in popularity. Each in contrasting neighborhoods.
Pubs are fabulous spots to engage in conversation. Some of my favorite pubs include the Red Lion down the Crown passage. Fun locating it. Tiny. The Blackfriar ( as already mentioned). The Princess of Prussia. As of November 2015 still had old world charm, i. e. had not become a wine bar or gastro pub.
The Castle in Holland Park, www.castlehollandpark.co.uk, discovered and appreciated on my last London visit. Was close to the AirBnB apartment we rented. The Mayflower in Rotherhithe. Enjoy the fireplace in March. The Duke of Hamilton which the Hampstead Rugby club calls home.
Try to participate in an evensong at St. Brides church. That will put you in contact with the "locals."
As far as non touristy sites, Liberty department store, Alexandria Palace ( check the website to see whats happening) the rolling bridge in Paddington basin at noon on Fridays. The Camley Street Natural Park adjacent to Kings Cross St Pancras Station ( don't miss seeing the statue in St Pancras Station, trust me, you can't miss it and St Pancras is gorgeous), the Whitechapel Bell Foundry, the roof gardens at Kensington High Street, see whats on at Wilton's Music Hall, St. Dunstan in the East Church Garden, visit the Hindu Temple near Wembley park are some thoughts.
For must do options book the Tower of London ceremony of the keys for one of your 4 nights, probably on the same day you've toured the Tower. Visit the Treasure Room at the British Library, book a tour of the Houses of Parliament, see Westminster Abbey, lunch in the St Martins of the Field crypt cafe, Churchill War Rooms, walk along the Thames path beneath the London Eye, provides a lovely POV of the Houses of Parliament as sun sets. Lastly, would take up most of a day but Hampton Court is wonderful.
Read the RS guide to London, check TimeOut London online to see what will be on ( art, theatre, events, etc) while you are there, buy the Oyster Card when you are there, and off you go.
