In a large pot over medium-high heat, bring 1/2 cup of the broth to a simmer. Add the onion, carrots, and celery and cook about 8 minutes, or until onion is translucent, stirring occasionally.
Add the remaining broth, along with the water, rice, chicken, and bay leaf. Bring to a boil. Reduce the heat to a simmer, cover, and cook about 35 minutes or until rice is tender and chicken is cooked through.
Remove the bay leaf and stir in kale. Continue cooking just until kale is wilted and tender, 3 to 5 minutes.