A couple of suggestions from an Inner City kid that are not in central London.
The Museum of Childhood in Bethnal Green was fascinating for my GD when she was about 6. Lots of old games, dolls, clothes and loads of other things to see. It was also great for me as I saw some stuff there that I had played with as a child! It's very near the tube station and easy to get to.
In Shoreditch there is the Museum of the Home, which has various room settings from different ages. We used to visit it from Junior School (by foot!) so it may not be as good for a 5 year old, but it is very child-friendly - and free!
I agree with the Coram's Fields suggestion if you want a safe, secure park. Adults are only allowed if accompanied by a child and it is a lovely space for them to run around and explore.
Speculation: Who Will Be In Burnham's Cabinet?
