Many of the museums are free, so that helps. Imperial War Museum, National History Museum etc.
Accom is a killer, so you have to look at some cheap options. Going around by Tube is not that cheap also. If you can walk a lot, it helps.
Food-wise, eat cheaply. We like Greggs (even cheaper than M&S and Pret A Manger), but they are not always available.
See the video for our Greggs experience in London.
This video on Imperial War Museum, which my kids loved.
Oh yes, some pubs serve decent food at prices that are not too bad. We like the Wetherspoon's chain. Here are some video examples of our experience.
Brought the kids to eat here too.
This lunch cost me 8 pounds. There were cheaper options.