I can't tell you which properties would give you free breakfast over and above standard benefit (which is none as you know), but if I had two expiring free night certs, I wouldn't really be equating a HIX with an IC. There's just no comparison. I find it hard to eat the breakfast at the HIX at any rate, but that may be me.
However, if you are going to insist on paying for breakfast if it isn't available for free, then the IC will be very pricey. I would suggest going to a Sainsbury's local or whatever and spending GBP5 on breakfast goods/ fruit etc instead.
In terms of location: I love wandering around Mayfair when in London, so again, the IC is excellently located. But this is a first visit, in which case it's also fairly close to the tube.
Just to tell you what you already know: cramming too much in with a young child will not make for happy times, and you are only there a couple of days. Despite what will be gigantic queues, if your child might appreciate the dinosaurs, the Natural History museum is a favourite, and will be far more interesting for them than Buckingham Palace, the Tower of London etc etc.
Enjoy!
tb