Most Fish & Game Commissions are ruling Mute Swans as non native and nuisances. They may help you capture it as well as the rest of the family in order to kill (cull) the swans in order to control their population. So you need to check their stance on the issue prior to seeking their help.

You might check with a non profit wildlife/waterfowl rescue or rehabilitation center for help. If the cygnet is not treated in a timely manner, the break if treatable will not heal properly. The bird will also be susceptible to predation. If the break is not treatable, depending on the location of the break, the wing may need to be amputated. Unless it is treatable and even then , the prognosis is slim in returning the cygnet back to the wild any time soon or even at all. It may live a great life in a very controlled captive setting as long as it sheltered from predators, inclement weathet and provided with supplemental food. The Regal Swan