Apple has held mixed views on broken hinges. I'd absolutely take the computer to your Apple Store (if one is near you) and see if it's eligible for warranty service. If it's a "covered repair," you should get a quick (2-4 day, perhaps) screen replacement, free of charge! If they consider it "accidental damage," they should quote a price of $430 for Tier 2 Accidental Damage.
If they quote accidental damage pricing, definitely call up your local Apple-Authorized Service Providers. Be warned, they may not be able to give an exact price quote over the phone; this is not a common problem, and they may need to do some research before discovering what parts need replacement.
If nobody local will do this cost-effectively, you should find several mail-in service providers with competitive pricing. Parts, labor, and shipping should not be more than $250.