We've been to London about 4 times and are planning a short visit during a layover this summer.
In terms of getting into London, we recently came across this video by comedian Jay Foreman did for London Planner. It was posted in 2017 but some of the info is already out of date. He's done a few videos for London Planner they are amusing enough to check out.
Taxi's are obviously most expensive. The Heathrow Express is quick with no stops but also pretty pricey at 22GBP. In the video, he talks about the Heathrow Connect which has a few stops but is half the price of the Heathrow Express. However, it appears to have been bought and replaced by TfL Rail and renamed the Elizabeth Line, which actually might be a positive because it looks like you can use the Oyster card for it. Lastly, he talks about taking the Underground/metro which is even cheaper but takes an hour to get into town.
A few of our faves are:
Southbank walk from Westminster Abbey to Tower Bridge. You past by so many big sites: Big Ben, London Eye, The Globe Theatre, Borough Market, City Hall, etc.
Harrods Food Hall. The breadth of items here is amazing.
The area around Piccadily Circus, Chinatown, Leicester Square, and Covent Garden at night. It's just such a lively area.
National Portrait Gallery. We love the history around the British Monarchy. The National Portrait Gallery kind of gives a timeline of not only the monarch but influential people during the time like scientists, explorers, etc.
Tower of London and Ceremony of the Keys. The Tower of London is cool with the Crown Jewels. But you can also get tix to the Ceremony of the Keys which gets you in at night with a limited group.
We also really enjoyed trips out to Windsor Castle and the Royal Observatory.
During our last trip, we also visited the Brick Lane area which is kind of a hip revitalized area which has a Sunday market that may be fairly close to where the OP is staying.
If you have access to data on your phone, try using google maps to identify journeys by bus instead of taking the Underground. On our last trip, we rode double decker buses as much as possible and we obviously saw so much more by travelling street level.
We bought a London Pass during a previous trip but won't do it again. We kind of felt forced to see too many "B" attractions that we only found meh.