How old are your toddlers?
Another voice for an apartment ... we typically eat breakfast and dinner in the apartment and splurge on lunch out. Or, sadly, sometimes eat crappy-but-expensive take-away lunch on the go.
I have rented apartments through Home From Home and been very happy with their service and properties. I've rented a 1BR but you may prefer 2 ... so can't offer any scoop on a particular 2 BR flat. But in terms of neighborhoods, I will say that when we stayed in Holland Park, we were about a block from Holland Park itself, with its two playgrounds, which was fabulous. The toddler lot is a little small, but the bigger-kid playground worked great for my four-year-old, who loved the zip line. There is also a kid-friendly Giraffe restaurant and an ice cream store in the neighborhood, plus the second branch of Daunt's Books, which has a nice kids section.
Another time we rented a flat near Sloane Square. It worked great for us, but the main advantage of that neighborhood is easy access (by bus or Tube) to the Kensington museums, which may not be such a draw for you, depending on age. (Well, plus the grown-up shopping, etc of Chelsea).
Some of our successful London activities have been:
Playgrounds -- Holland Park, the Princess Diana playground in Kensington Gardens, Corams' Fields in Bloomsbury
London Transport Museum, followed by watching the buskers at Covent Garden
Boat ride to Greenwich, followed by trip to the naval museum there
Science Museum -- very interactive, good younger-kid play area in the basement
Daytrip to Hampton Court Palace -- they have (or at least had) people playing Tudor-era court characters, including Henry VIII, which my daughter found very exciting
Hackney City Farm (there are other city farms around too, but we do a full Sunday morning in the East End -- Columbia Road Flower Market early, before it gets super-crowded, followed by breakfast at the cafe at the farm)
Childrens' theater activities at the National Theatre -- we were there last summer during Children's Theatre week; many of the West End shows offer free tickets to kids, but the shows are still too mature for young kids. However, the National had some lovely performances appropriate to the younger crowd, plus a behind-the-scenes theater tour for kids and their families
I would NOT recommend the London Aquarium for the (steep) price; it's an ok exhibit but not worth that kind of money. It was also a long wait and quite crowded.