Mix chili powder, salt, and cumin together in a bowl. Rub mixture all over chicken.

Pour chicken broth into a multi-functional pressure cooker (such as Instant Pot). Add a rack to the bottom of the pot and place seasoned chicken on top. Close and lock the lid. Select high pressure according to the manufacturer's instructions; set the timer for 5 minutes. Allow 10 to 15 minutes for the pressure to build.

Meanwhile, preheat the oven to 350°F (175°C). Lightly grease a 9x13-inch baking dish.

Release the pressure using the natural-release method for 5 minutes, then quick-release the remaining pressure for 5 more minutes. Unlock and remove the lid.

Stir 2 cups Monterey Jack cheese, 1 can green enchilada sauce, green chilies, and onions together in a large mixing bowl to thoroughly blend.

Transfer cooked chicken to a cutting board and shred or chop into small pieces. Add chicken to the mixing bowl and stir until blended with cheese mixture.

Heat olive oil in a skillet. Soften tortillas, one at a time, by dipping into the hot oil for 2 to 3 seconds per side. Drain on paper towels.

Spoon 2 to 3 tablespoons chicken mixture onto each tortilla, then roll up and place seam-side down into the prepared baking dish.

Combine remaining green enchilada sauce with red enchilada sauce in a bowl; pour over enchiladas in the baking dish. Sprinkle remaining Monterey Jack cheese over top.