The vast majority of the museums are free - you just need to book a time slot.
Fountains by Kings Cross (Coal Drops Yard) - take a spare pair of clothes.
Golders Hill Zoo and Butterfly House are Free
Homerton Adverture Playground is Free
Street Art Walks are free and usually pretty fun
Camden Markets (Free, but food can be a but pricey)
Canary Wharf and South Bank are both good ideas (both have different things on)
Exhibition Road Festival is free (if it's on when you are here) - usually many street performers there too.
Street performers around Covent Garden as well.
You can visit the Langar and get free food in Willeseden - it's a beautiful space and very nice to introduce children to the religion and charity.
London Mithraeum, Barbican (not the exhibitions), but i think they do soft play which is (not sure if it's up and running again, they also usually do a Summer play festival, and its next to Museum of London.
Changing of the Guard - free
Pretty much all the markets are free to enter and not limited to capacities at the moment.
Lots of places do free musical events at lunch times.