Preheat the oven to 350°F (175°C).
Combine the shredded chicken, black beans, corn, and half of the enchilada sauce in a skillet over medium heat.
Mix the cheddar and monterey jack cheese in a mixing bowl.
Spread a thin layer of enchilada sauce on the bottom of a 9x13 inch baking dish.
Layer half of the tortilla strips over the sauce.
Add half of the chicken mixture on top of the tortillas.
Sprinkle with half of the cheese mixture.
Repeat the layers: tortillas, chicken mixture, and cheese.
Pour the remaining enchilada sauce over the top and sprinkle with any leftover cheese.
Cover the dish with aluminum foil and bake for 20 minutes.
Remove the foil and bake for an additional 10 minutes, or until the cheese is bubbly and golden.
Let cool for a few minutes before serving.