Preheat oven to 350°F. Spray a 9x13-inch baking dish with cooking spray.
In a large bowl, combine the chicken, lime juice, chili powder, onion powder, garlic powder, salt, pepper, and 1 cup shredded cheese. Mix well until all ingredients are evenly distributed.
Working one tortilla at a time, place about ½ cup of the chicken mixture onto each tortilla. Wrap tightly and place the filled tortillas seam-side down in the greased baking dish. Repeat with the remaining tortillas and set aside.
In a medium saucepan over medium heat, melt the butter. Add the garlic and cook for about 1 more minute. Add the flour, salt, and pepper, whisking together until the mixture turns golden brown (about 2 minutes).
Gradually whisk in the chicken broth. Add the cream cheese and sour cream. Continue to cook, stirring often until the mixture is smooth.
Add green chiles. Stir to combine.
Pour the white sauce evenly over the rolled enchiladas in the baking dish. Sprinkle the top with the remaining cup of cheese.
Bake for 25-30 minutes, or until the cheese is hot and bubbly. (Broil the enchiladas for an extra 2-3 minutes if you prefer the cheese to be golden brown.)
Garnish the enchiladas with chopped cilantro. Serve with lime wedges on the side.