I'm sorry to hear about your duck's bill.
My duck got a similar bill injury and we took her to an avian vet. They weren't very concerned. They wanted me to put antibiotic cream (like Neosporin without pain reliever), just a thin layer on the wound twice daily for about 3 weeks. They told me to keep it clean, and change her water and clean her bowls/buckets frequently.
My duck healed slowly, but when I brought her in for a check-up they thought she looked great. They also told me they thought she stuck her bill through the wires on her run/aviary and scraped it coming back in. Maybe that's what happened to your duck.