I had one thrown out of the nest like this too, years ago. We were never sure why the hen threw it out, and I hatched it myself. It seemed fine, but died about a week later from eating something bad -we had wood pellets around the brooder we were keeping it in, and we expect it thought these were food and died from eating them, as did the other duckling we had. It was very sad.
Anyway, that egg needs to be warmed up -put in an incubator, or even put under a lamp will do- and then you need to check the yolk. Is it absorbed? has it been punctured? In either case, you can probably remove the rest of the shell from the duckling at this point as long as it does not cause bleeding, as that duck will never be able to "pip and zip" with its leg hanging out like that. That will also help to ensure a clear airway for the duckling.
If the yolk has been punctured -I had a duck like this once, she was pecked out of her shell by chickens and they pecked the yolk too- you'll need to remove the remains of it, stop the bleeding, and make sure the duck's intestines are in the body and remain there. With mine, all I had to do for this was remove the yolk and stop the bleeding, then place the bit of intestine that was outside -looked a bit like a looped earthworm- back in through the hole in the belly and hold it there for a few minutes. I then put the duck back in its incubator to rest, and began offering food and water the next day to make up for the lack of yolk. That duck survived to grow into an adult duck, and I hope yours does too.
How is it doing now?
