Any farms nearby actually now that I think about it runoff with fertilizers and pesticides in it could be getting into the pond and making you duck sick.
Diluted peroxide and any pictures? Glad to hear the chick is ok. I recently lost my chickens to a coyote. But we got some new ones. But pictures would be greatly appreciated.
Ducks love mud... and they love to eat... Frogs... Idk why but mine ate any frog brave enough to enter their pool. Clean their area a lot because poop will just build up and be gross.
Try to feed it foods high in protein like finely chopped hard-boiled egg. Try putting a drop or two of sugar water on its beak to give it energy. Too much sugar will give it diahrea.