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Disclaimer: This is in regards to birds with a hard beak such as chickens, parrots, etc. Ducks are a different matter!
You are close! A bird's beak is an extension of their jaw, and is bone. On top of the bone the beak is covered in keratin (same material as our fingernails). The keratin sheath is called the rhamphotheca and is (slowly) constantly growing! The portion of the beak closest to the base of the beak has a blood supply and nerves. It is understood that the tip of the beak (in species with a hard beak) does not feel pain as it has no nerve endings, but as you get closer to the base yes! They absolutely seem to feel pain. If this area of the beak gets damaged, it may not be able to regenerate the tissue and rhamphotheca again and the beak will not regrow (for parrots and other companion birds some people get prosthetic beaks made of acrylic). In the practice of 'debeaking'* chicks (usually for factory farming on a large scale of chickens for food and/or eggs) the chicks are subjected to damage to the nerve/blood supply of the beak so that the beak will not regrow properly (sadly it is to keep them from damaging one another in overcrowded factory situations).
So it all depends, when a beak is injured, on how it was injured and where! It may or may not grow back with time.
* Not to be confused with clipping an overgrown beak, which only involves trimming away the nerve-free tip, at the very end, if the beak overgrows (like a fingernail).
Mystus808, I'm sorry I don't have any further advice... I am not real clear on how the beak broke, sounds scary! Any chance you can get a photo for us?
