Good restaurant around frenchmen street after midnight
13 will be open and it's really good and really close to you. The Verti Marte is nearby and if you flake out they deliver. I don't recommend La Peniche. The last late night breakfast there will remain exactly that -- the last.
1940 Dauphine St, New Orleans, LA 70116
1201 Royal St, New Orleans, LA
La Peniche (1940 Dauphine, corner Burgundy) is four blocks from your lodgings and open 24 hours on the weekend. It's nothing fancy but fine for breakfast items or sandwiches. Basically it's diner food in slightly nicer surroundings.
Yuki Izakaya is a couple doors down from where you're staying, and they serve great Japanese until 2 or 3.