Place the oil, onion, carrots, kale, and celery in the Instant Pot, along with the salt and pepper. Turn on the "saute" function and cook for a few minutes, stirring with a wooden spoon.
Turn off the saute feature. Then, add the chicken, 4 cups of chicken broth, and 4 cups of water.
Seal the lid and set the timer for 25 minutes.
Release the pressure when it's finished cooking. Then, transfer the chicken from the pot to a cutting board. Remove the meat from the bones and add the meat back into the pot.
Stir the soup to recombine the ingredients. Serve and enjoy!