The restaurant in the Trump hotel is supposed to have a great kobe burger, as well as the Oprah rave for their turkey burger.
Midnight in that area is going to be tough. Most places are closed by 10 or 11.
Tempo Cafe, at State and Chestnut, is within walking distance and is open 24 hours. Good food, not sure about their burgers, but probably fine.
6 E Chestnut St
Chicago, IL 60611