I don't know if that's a reasonable rate for London, sorry! I have never had to rent here. Balham is not a bad area, but it is a little far out of the centre and it's on the Northern line, which in this weather is pretty horrible! London is absoutely roasting at the moment. Of course, it might not be by mid-July, this being the UK!
I would say that Balham would be a bit of a pain to get to from Stansted. You'd take the train to Liverpool Street (there's not really a better way of getting to central London from Stansted) and then walk to either Moorgate or Bank to get the tube. There are other routes, but that would involve the least changes. It's still going to take you 90 minutes to get from the airport to Balham.
Have you looked at flights into City Airport? It's a much nicer airport and has better travel links.
In my view, things not to miss are:
the South Bank for walking, people watching and lots of free attractions;
Tate Modern and Tate Britain (both free)
V&A
Science Museum
A boat trip on the river
Greenwich for the maritime museum, cutty sark and the observatory
Houses of Parliament
Covent Garden is fun for mooching around.
Avoid Oxford Street
Harrods is a waste of time and money
The Elizabeth tower/ Big Ben is currently covered in scaffolding and plastic sheeting whilst they renovate it so you won't get to see or hear it, unfortunately, but the Houses of Parliament are lovely to look at (though bits also covered in scaffolding).
On one of your days I would recommend taking a tube to Westminster, walking past the HoP down to Tate Britain. Mooch around TB then catch a Thames Clipper boat to Tate Modern, then walk back from Tate Modern along the South Bank. Or do that in reverse!
I never pay to get in anywhere so can't comment on paid for things!
I hope you have a wonderful time. I love my city and hope others do as well. Inbox me if you want more info.
Hopefully people with younger kids can give you an idea of things to do that would work for you all as a family.