Preheat your oven to 350°F (175°C) and lightly grease a 9x13-inch baking dish with non-stick spray.
In a mixing bowl, combine 1/2 of the cream cheese, chicken, 1 cup of cheese, lime juice, cumin, and chili powder. Season with salt and pepper if desired.
Take a tortilla and place 2-3 tablespoons of the chicken mixture on it, then roll it up with the seam side down and put it in the prepared baking dish. Repeat this process with the remaining tortillas and filling.
In a medium saucepan over medium-high heat, melt the butter. Stir in the flour and cook for 2-3 minutes, or until it turns a golden brown color. Season with salt and pepper.
Gradually whisk in the chicken broth and green chiles. Add the remaining 1/2 of the cream cheese and cook for 5-7 minutes, or until the sauce thickens.
Remove the sauce from the heat and pour it over the enchiladas in the baking dish.
Top the enchiladas with the remaining cheese.
Place the baking dish in the preheated oven and bake for 22-25 minutes, or until the cheese and filling are melted and bubbly.
If desired, you can broil for 3-5 minutes to lightly brown the top.
Remove from the oven and allow it to rest for 10-12 minutes.
Optionally, garnish with fresh cilantro before serving.