If you're visiting for a week or longer you may want to consider looking for a holiday flat (self-catering) instead of a hotel room. I know they're more self-sufficient, but it would allow you to stay as a family (and in more of a family environment) while you're in London and potentially cost you less than having multiple hotel rooms... My in-laws usually rent one when they go on holiday (mind you they don't go to London anymore since my SIL moved back up north), but when they go away thats what they prefer, so that they can bring the dogs...
Just a thought or something to look into. I'm sorry I don't have any booking site recommendations as we've never tried it for London, but I'm sure if you did a google search or check
Visit Britain (it has an accommodation checker) or
Enjoy England you'd be able to come up with something more useful.
Good luck!