Remove skin from chicken. Trim off and discard excess fat. Rinse chicken and pat dry. Rub chili powder all over pieces.
Set a 12-inch frying pan with 2-inch-tall sides or a 5- to 6-quart pan over medium-high heat. When hot, add oil and tilt pan to coat bottom. Add chicken in a single layer and turn as needed to brown on both sides, 10 to 12 minutes total (if necessary, brown in two batches, adding 1 more tablespoon oil if needed). Transfer to a plate.
Add onion, garlic, and jalapeño chiles to pan; stir often until onion is limp, about 3 minutes. Reduce heat to medium and add rice; stir often until rice is opaque, about 3 minutes. Add broth, tomatoes (including juices), corn, and 1/4 cup mint. Bring to a boil over high heat.
Lay chicken pieces slightly apart in pan. Reduce heat to low, cover pan, and cook until rice is tender to bite and chicken is no longer pink at the bone (cut to test), 25 to 30 minutes. Sprinkle with remaining mint and add salt to taste.