Let's see, Conch Key is probably going to be the quietest. The only foot traffic in that area is from Narcoossee's restaurant and the boat dock. It is the furthest from the pools and the main building though. I'm talking maybe a 2-3 minute walk at the most.
Sago Cay is probably the most private. It's kind of set off to the side by itself and the front overlooks the marina, while the back has the gardens and monorail going by. That might be kind of cool for kids to see. It's the closest to the main building.
Big Pine Key and Boca Chica will put you right next to the center courtyard pool. If you're looking for quiet as you sit on your balcony, they might not be the best choice, but they are extremely convenient to the pools and the main building.
I hope that helps.
Tammi
