Preheat oven to 350 degrees F.
Coat a square baking pan with cooking spray.
In a large bowl, combine chicken, sour cream, 3/4 cup of the Monterey jack cheese, 3/4 cup of the cheddar, salsa, chiles, chili powder, and cayenne pepper. Mix well and set aside.
Arrange half of the tortilla pieces in the bottom of prepared pan, overlapping pieces slightly to cover the surface. Top with half of the chicken mixture and smooth over with the back of a spoon to even the top. Layer remaining tortillas over top, and spoon over remaining chicken mixture. Top with remaining 1/4 cup each of Monterey jack and cheddar cheeses.
Cover with foil and bake 30 minutes. Uncover and bake 30 more minutes, until top is golden and bubbly. Let stand 5 minutes before serving.