Preheat the oven to 350°F (175°C).
Heat olive oil in a large skillet over medium heat. Add diced onion and green bell pepper and cook until softened, about 5 minutes.
Add minced garlic and cook for 1 minute until fragrant.
In a large mixing bowl, combine the cooked chicken, sautéed onion, bell pepper, and garlic, diced tomatoes with green chilies (including liquid), cream of mushroom soup, cream of chicken soup, sour cream, chili powder, ground cumin, salt, and black pepper. Stir until well combined.
In a 9×13 inch baking dish, spread a thin layer of the chicken mixture on the bottom.
Layer one-third of the tortilla strips over the chicken mixture.
Sprinkle one cup of shredded cheddar cheese and one-third of the Monterey Jack cheese over the tortillas.
Repeat layering two more times: chicken mixture, tortilla strips, and cheeses, ending with cheese on top.
Cover the casserole with aluminum foil and bake for 30 minutes.
Remove the foil and bake for an additional 10 minutes, or until the cheese is bubbly and golden brown.
Let the casserole rest for 5 minutes before serving.