If you go to St Paul's, the Museum of London is only 5 minutes walk away, and the Bank of England museum is about 10 - 15 minutes walk away.
Harrods does have a dress code (haven't been for years so I double checked their website to see if it had changed), nothing major though, no visible tattoo, no offensive slogans on clothes and if you are wearing a rucksack you must be carry it in your hands or in front of you.
Personally I don't like the Museum of Childhood, it would bore my almost 10 year old, but all of mine went to the Museum when they were 5 on school trips (so it could be my experience as a parent helper trying to keep track of 4 Y1s around the Museum and on the train which could have clouded my opinion)
Kensington palace is good and you can get a joint ticket with The Tower of London.
Something I want to do is Hampton Court palace, one day I will get there.
Enjoy your visit and try not to worry about what is happening. I worked in London through the IRA bombings, they even blew up my office in one of the bombs, and never really though about it.
It did feel a lot different being up in London in July 2005 (about a week after 7/7) with a 3 month old DC1 when there was a second bomb scare. I managed to walk him across town to get to mainline station after the tubes were suspended with the help of my trusty A - Z.