London is one of my favorite global cities to visit. I hope you enjoy your travels. I'm jealous!
If I had limited time, then my top places to go would be the British Museum and the Tower of London.
Piccadilly Circus is a great place to people-watch especially on a weekend night. There are also loads of restaurants to eat in the area.
If you look at a map, it's not that far to do a walk from Piccadilly Circus past the shops on Piccadilly down to Green Park. It's a lovely walk through Green Park past Buckingham Palace and then along St James's Park on Birdcage Walk down to Westminster Bridge where you can see Westminster Abbey, Parliament, Big Ben, and also the London Eye on the other side of the river. It's worth it to walk onto or even across the bridge, and you can get some beautiful photos. After that you can walk down Parliament St / Whitehall to the Horse Guards Parade. That takes you right past Downing Street which is gated (and heavily guarded obviously!) but at least you see the area where the Prime Minister lives. Keep following the road and it takes you to Trafalgar Square. From there you can walk up to Leicester Square and back to Piccadilly Circus. You could modify this walk to make it shorter. Or, you could spend a whole day just leisurely exploring this area of town if the weather is nice, and you would see many of the famous sites.
One of my favorite things to do is go to the theaters in the West End. You can buy half-price discount tickets in Leicester Square.
There's also the National Gallery at Trafalgar Square. There's a Sherlock Holmes Pub on one of the side streets from the square. Food is just mediocre, but it's a novelty.
http://www.sherlockholmespub.com/history.php
No one has mentioned St. Paul's Cathedral and the Millennium Bridge.
I've always wanted to visit the Victoria & Albert Museum but never had time.
Also, many museums have a cafeteria inside with fairly decent (although expensive) food. Sometimes that's a time saving tip on a busy day.