I love the British Museum, been on many school trips there as a parent helper. Looking at the crystal skull or seeing an Easter Island Head used to fascinate the groups of kids I was trusted with, also stopping to speak to the volunteers who let the children handle artefacts. The teachers were happy as long as I took them to the Mummys on the top floor which was why we were there.
The British Museum is close to Forbidden Planet too as it is near Holborn instead of South Ken.
I have also been on school trips to The Museum of Childhood in Bethnal Green with the Reception class and it took a lot to keep their interest for the whole day, so it is probably just a half day trip max for younger children.
As I live in an outer London Borough we have been all over with both our three children and school trips. While we enjoy the Natural History Museum it is just a crowded, we had annual tickets to the Science Museum interactive area and that is really worth it.
The Tower of London is good too, but it would depend on the time of year and what is on. They used to have dressed up performers telling history, My kids loved and we had annual tickets for years, never really bothered with the massive queue to the Crown Jewels.
I found they got tired easily, so do short bursts and then sit down or play before you move on to the next activity.
I love travelling around on the top deck of the buses and looking at statues and carvings in the buildings and reading all the blue plaques.
If you ride the DLR and end up in Stratford, the Olympic park is a nice area to decompress away from the bustle of the City, but don't go on a West Ham Home match day.
I would agree with City Garden over the Shard, have horrid memories of spending ages trying to occupy three bored children at the top of the Shard while DH spend ages looking at the view. He didn't return the favour either for me to have a look.
The City Garden is also very close to Leadenhall market which was used was a location for Diagon Alley in the Harry Potter films
Have a lovely visit, London is a beautiful and diverse City, no matter how many times you have been there, there is always something new to discover.