Seems to make sense to set the patio adjacent to the kitchen area.

I'm in agreement with Beverly that the general obvious location would be at the upper left of the house. Often, fire pits are some distance from the house, if not separate from the patio. Projecting the patio beyond the end of the house would give you visual access to the play area. Some of the patio may abut the house walls, or it may not if you want to allow space for foundation planting between house and patio. Exact dimensions, shape and all the details is just something you'd work out in the design process.

Yeah but if you tuck the patio in the obvious spot (the corner) you can't see the swingset, so it's almost like disregarding one of the design requirements. There's no reason the patio has to directly abut the house. Slide the whole thing out so you have a clear sightline. It gives you great planting beds, and eliminates the problem of that patio edge along the house that collects leaves and crap all winter long, stains, and is generally unused anyhow. As far as the kitchen passthrough, I'd think long and hard on that. We've designed around that concept for 5 or 6 clients over the years, and precisely zero use the passthrough. I guess it's just not as convenient as it sounds.