Hi Magsie. if you type in Kid's London there is a site with family friendly accommodation which looks promising.
London is expensive but there is lots of free stuff. The science museum is fantastic,and the cafe there is good. But I would say get there when it opens as it is , as all attractions are, rammed.The National Gallery in the holidays usually has a programme for kids, and it has a fabulous restaurant/cafe/bar above the Sainsbury Wing-very grandparent friendly with delicious cakes good food, and wine! It is child friendly too! A walk along the south bank past the national theatre is always good in the summer. They do kid friendly things all around that area. There are lots of places to eat, such as Giraffe, but I would reccommend Canteen behind I think, the National Theatre. HMS Belfast is free for kids.
Not free but worth doing , is the Duck Tour, for the last bit it goes in the water in front The Houses of Parliament. Kids love going round the Covent Garden area, there is always lots going on there. I don,t know how much it is anymore, but I used to take young visitors on one of the open top buses, they take you all over central London and you can get on and off when you want. The London parks are lovely, and in Kensington Gardens, or it might be just in Hyde park, there is the Princess Diana Memorial adventure playground. Near there you can refresh yourelf with tea coffee and cakes in the Orangery, which is beautiful.