In a mixing bowl, add the shredded chicken, half the sour cream, and half of the cream cheese. Mix well to combine. Add half of the lime juice, stir, then set aside.
In the bowl of a food processor, add the whole jalapenos, green chilies, and the remaining sour cream. Pulse for 5-10 seconds, until chopped but still chunky. Add mixture to the bowl of chicken and stir well.
Layout the tortillas. Spoon 2 heaping tablespoons of the chicken mixture in the center of each tortilla and then top with 1 cup Monterey jack cheese. Roll up each tortilla and place seam side down in a baking dish.
Put into oven for 10 minutes, or until edges are golden brown.
While cooking the enchiladas, make the sauce. Melt butter in a small saucepan, over low heat. Once melted, whisk in the flour. Let cook for 1 minute. Add the milk and increase the heat to medium. Once the milk boils, reduce heat to a simmer and let thicken, 1-2 minutes. The sauce should coat the back of a wooden spoon. Add the remainder of the cream cheese and stir until melted. Add the other 1 cup of Monterey jack cheese and stir until melted. Add the sliced jalapenos. Taste and add salt and pepper, if desired.
When the enchiladas are cooked, top with the sauce. Return to the oven for another 10 minutes, or until sauce is bubbling. Serve immediately.