I've lived in London, various places within the M25, for about 40 years now and still walk around looking up and enjoying exploring both new and old places. We do the Hidden City treasure hunts which is a great way of finding new spots.

Tower of London is good, there is so much history in there and a number of exhibitions well worth seeing.

I've done most of the "tourist traps"[1] and really enjoyed them. The London Eye gives you great views, especially at sunset; The Shard (cocktail bar, not done the viewing platform) is a bit expensive and style over substance but still worth doing once; Buckingham Palace is a world class historical living building; The South Bank has a lot going on and also gives great views of London; The Royal Albert Hall is a stunning concert venue; most of the big museums are free.

The one place I've not done as it really doesn't appeal to me is Madame Tussauds.

[1] Compared to other cites apart from the pedal rickshaws and the find-the-lady on Westminster bridge I don't think London is inundated with rip-off tourist traps like many other cities appear to be.