"Also, are hotels pretty strict about the number of people in a room? For instance, would it be ok to book a triple room with 3 adults and 1 kid?"
Yes, they are strict, and no, you won't be allowed to book a triple and put in four people. When you see the size of the rooms in London (or, for that matter, any other European city), you'll see why. In addition to violating fire laws, there's usually no way an extra person (even a small one) can fit; the beds often take up much of the (small) room. Furthermore, beds are usually smaller in Europe; rooms don't have king or even queen beds, but usually double beds (of course there are exceptions). A triple room will often have a double and a single bed (think cot width - only twin width if you're lucky).
If a hotel looks promising but you don't see quad rooms on the automated booking engine, e-mail them directly to ask. Many hotels don't show all the rooms on the booking engines (even the ones on their own websites). In your e-mail, be specific about what you are looking for. For example, "We need accommodation for 5 adults and 3 children, ages x, y, and z. Do you have rooms for us, and what would the total cost be?"
Have you looked at the Premier Inn? They are a chain and have family rooms (be sure to select the number of children and adults correctly). You will need two rooms and this might be over your budget, but it's worth checking for your dates: http://www.premierinn.com/gb/en/home.html.
If you find one that has a suitable price, ask here about the neighborhood.
If the Premier Inn is too expensive, here's a listing from EuroCheapo. Again, in your situation I'd contact them directly via e-mail. http://www.eurocheapo.com/london/
