I can empathize with your situation. My first attempt at killing a chicken left me feeling horrible because I don't think I cut the neck deep enough, meaning the chicken died, but not as quickly as I would have liked. I think the mistake I made was that I had to come to grips with the fact that while I could be gentle with my chickens when I was raising and handling them, and I could be sure that all their needs were met and that they were subjected to as little daily trauma and stress as possible and even when I was carrying them to the killing cone and putting them inside, I could be calm and gentle, but that there is no "gentle" way to kill. Trying to be gentle at that stage is the wrong thing, then you need to be quick and decisive. You need a sharp knife, you need to hold the duck's head in your non-knife hand and then place the knife close to the duck's neck and move it towards the head so that you push the feathers asside and get the blade up against the skin because otherwise you're trying to cut through the feathers, and they act like armor to protect the bird's skin. Then you need to slice deeply from more or less under the ear to more or less the front of the throat.
I do another process first, called pithing or debraining, where I (gently) open my chicken's beak and not gently stick the point of my boning knife up through the roof of the mouth into the brain, so the brain is dead before I even make the cut to the throat, making me feel better that the chicken is already beyond feeling anything if I don't do everything perfectly. I've never butchered a duck and would suggest doing some web searching on the subject if that method might be easier for you.
Good luck, and if you need an expert to help you the first time, that's OK too.