Heat the oven to 375°F and spread 1 cup enchilada sauce in a 9-by-13-inch baking dish.
Mix the cooked chicken, black beans, green chiles, cumin, ancho chile powder and 1 cup cheddar.
Divide the filling among the warm tortillas, roll them snugly and arrange them seam-side down in the dish.
Pour the remaining enchilada sauce over the rolls and top with the Monterey Jack and remaining cheddar.
Cover with foil and bake for 20 minutes, then uncover and bake until bubbling, 8 to 10 minutes.
Rest the enchiladas for 5 minutes before serving.