The St. Paul's Hostel, right near three different Tube lines, and several buses, can be booked by the room. And you're a bit off-season so you could probably have your choice at the moment.
It's the former choirboy school, right near the Cathedral, and the staff is very reliable (last summer, when I needed a place to which to forward something valuable while traveling, they were the place I thought of first, and they came through just fine.)
I stay there whenever in London; it's easy enough to reach KC/St.P. for the BL and almost everything else--the Exchange is within walking distance (well, I walk a lot, but it's not far) also.
The rooms have 2-3-4 beds--some very interestingly layered, like trundles but sort of modern-stacked--if your child is old enough, they might enjoy one of those.
You don't have to be a member of the IYH group, although rates are discounted for those who are, and they serve a very reasonable breakfast (cafeteria-style, with many choices: youguhrts, fruits, stewed tomatoes, eggs, toast, bacon/sausage etc.etc.etc. for not very much additional in the AM as well.)
See:
http://www.yha.org.uk/find-accommodation/london/hostels/london-st-pauls/index.aspxFeed the birds for me...!
