When in London you will still see double decker buses, phone booths and corner post boxes all painted that bright red color you hoped for. The British Museum has not yet lost its charm. The National Gallery is still in Cole Porter's manner of speaking, The Top, even with the modern wing that some love and others love to hate. The traffic still swirls round Eros in Piccadilly Circus, where you will not find any animals, but perhaps a few clowns nonetheless. Wellington still gazes out heroically from atop his Column in Trafalgar and the photogenic pigeons are ever present there.
The Tower of London is worth a visit no matter what A. A. Gill says, and if you want to see the Crown Jewels (no that does not mean Philip and Charles in their Balmoral attire on a windy day, but rather a lovely display of artifacts of a pricey nature that you can slip past on a moving sidewalk that would make Walt Disney proud), by all means go to the Tower. And if you want to go round again, you are welcome to do so, especially if the place is not overcrowded. Westminster Abbey is usually packed unless you are there out of season but go anyway and inch past the tombs of Good Queen Bess and Mary Queen of Scots the star crossed cousins who are entombed inches from one another. History is ever-present at The Abbey.
If you want a marvelous view of London, yes indeed the London Eye is the place; the price is dear but so is the view. If you are on a budget, and who is not, then go to Westminster Cathedral (NOT Abbey) and take the 100+ year old lift up to the top of the tower for the second-best view in London, it is very inexpensive and while there you can see London's flagship Roman Catholic Cathedral and ponder the wonders of striped Late Victorian Architecture at its best--the Cathedral is still being fitted out inside and if you have a pot of gold, you might just wish to provide the funds to decorate one of the side chapels. I am sure they will be very glad you did.
You can usually tour Buckingham Palace, such tours being an innovation begun after the dreadful fire at Windsor during the Queen's horrible year; and if you have a sharp eye you may even see where the paneling opens to reveal a hidden doorway. If you are asked to tea at Buck House as was my aunt, you can expect to be served those little Swedish butter cookies to "go with", right from the tin. Now you know.
Another Royal residence you can tour is Kensington Palace in its ever so pretty grounds. Kensington is a kind of Royal condo, for those lessor lights of the House of Windsor to have a roof over their heads, but it also has changing exhibits of some interest.
We saw Princess Diana's gowns on one occasion, and other elegant attire worn by various crowned heads throughout the 20th century. You can also see the Princess Diana memorial fountain, very moving, and follow the Princess Diana walk, and if you feel you must go to Harrod's then by all means, but really it is one of those places that once seen is quite enough. I suppose in the days before shopping malls it was more of an attraction than it is now.
Yes, take in a live show in one of London's diminutive theatres, you will feel close to the action no matter where your seats happen to be. Yes, ride the tube everywhere to get near where you are going and then stroll around. Yes, take a ride on the top of a double decker bus. Yes, take a ride on a boat in the Thames. Yes, see the Tate, the Tate Modern, The National Gallery and Sir John Soane's House. Yes, take a nap on your unfolded map of London in the middle of Hyde Park, what a peaceful respite in the midst of a busy visit.
There is more of my London to come...