In a large mixing bowl, combine the shredded chicken, Monterey Jack cheese, cheddar cheese, salsa, diced green chilies, ground cumin, chili powder, garlic powder, salt, and pepper. Mix until well combined.
Wrap the tortillas in a damp paper towel and microwave for 20-30 seconds, until soft and pliable.
Spoon some of the chicken mixture onto the center of a tortilla. Fold the bottom edge of the tortilla up over the filling, then fold in the sides and roll the tortilla up to form a tight cylinder. Place each rolled tortilla seam-side down in the air fryer basket. Spray each with cooking spray to help crisp the exterior.
Air fry at 400°F (200°C) for 12 minutes.
Serve the chimichangas hot, topped with your desired toppings.