Mix chili powder, salt, and cumin together in a bowl. Rub mixture over the chicken.
Pour chicken broth into a multi-functional pressure cooker (such as Instant Pot(R)). 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 manufacturer's instructions; set timer for 5 minutes. Allow 10 to 15 minutes for pressure to build.
Combine 2 cups Monterey Jack cheese, 1 can green enchilada sauce, green chiles, onions, and jalapenos in a large mixing bowl. Stir to thoroughly blend.
Preheat the oven to 350 degrees F (175 degrees C). Lightly grease a baking dish.
Release pressure using the natural-release method according to manufacturer's instructions for 5 minutes. Release remaining pressure carefully using the quick-release method according to manufacturer's instructions, about 5 minutes. Unlock and remove the lid.
Transfer cooked chicken to a cutting board and shred or chop into small pieces. Add to the mixing bowl with the cheese mixture. Stir until chicken mixture is blended.
Heat olive oil in a skillet. Soften tortillas 1 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 a tortilla, roll up, and place seam-side down into the prepared baking dish. Repeat with remaining tortillas and mixture.
Combine 1 can green enchilada sauce and red enchilada sauce in a bowl and pour over enchiladas in the baking dish. Sprinkle remaining 1 cup Monterey Jack cheese on top.
