I would take look on https://all.accor.com/ use the filters set your price parameter , number of guests (2 adults and I child) etc. You should see some of the mid to budget range places with availability in decent locations. look for the Ibis styles, Novotel , Mercure brands . Make sure to join their loyalty club to get genuine discounts. Premier Inns are also good. You should find somewhere within your budget.

AirBnb would likely work out cheaper when you factor in the ability to cook yourself and/or order in. Eating out in London can very expensive!

As for location, pretty much anywhere in central London inside, or close to the Circle Line would be good. Some areas are better than others and be aware that AirBnb owners can get a bit creative with their locations!

You are never far from a tube station but don't forget about the buses. They can sometimes be quicker, more convenient and you get to see more of the city when travelling around.

Check out the YHA London St Pancras hostel.

Yes, hostel.

You can get a private room, family, en-suite (restroom just for you and in your room), and only a couple minutes from Kings' Cross. It runs 119-169GBP a night, the staff is very nice, the breakfast is quite good and we stay there every time we go to London.

We are also a family of 3 traveling July 3-9. Having the same issue. Here are a couple of places I am considering...I have never been to London so have no idea other than reviews Ive read online if these are good places to stay (maybe others could advise) They are both £338 per night. The Residence Inn also includes breakfast. Just thought Id share what I have been able to find.
Marlin Apartments St Pauls (one bedroom apartment)
Marriotts Residence Inn Tower Bridge (one room open plan with breakfast)

I would normally recommend my favorite family run Celtic Hotel, just off Russell Square and short block to Piccadilly Line. Perfect neighborhood. They have triples, and lovely breakfasts. However, if you hit an unusual heat wave they do not have AC.