Well, I've braved the Tube with a buggy and my dh in a wheelchair and I have to say it was so unbelievably awful that 'just' a child in a buggy, luggage and an older child would seem like a walk in the park in comparison (Not really - and we had an extra pair of hands to help as well, thank god).
First off, I would second the bus recommendation. I think there's a direct one from Kings X/St Pancras.
By tube, both Kings X and Hammersmith are 'wheelchair friendly' (i.e. lifts to platform level) but not on all lines. I did a quick search on the TFL site and it looks as if you could get on the Hammersmith and City line and go straight through, which would be relatively simple.
The main hassle is you have to wait for someone to let you through the ticket barrier and then go hunting for the lift. One of them, potentially at Kings X, I can't remember now, is bloody miles away and in a really weird location.
As the others have said, if you're not traveling at rush hour, you should be fine. Expect it to take some time but at least you don't have the problem that the allegedly wheelchair accessible stations still expect you (somehow) to be able to push a wheelchair up a step on to the train itself. They're not level with the platform or anything as advanced as that.