Well first of all you'll need a replacement hinge. They show up often on eBay, because they get broken often. Apple charge alot for them, so they aren't that cheap on eBay either. Make sure you get the correct one.
This is quite a big job as it happens, I swapped out some hinges on this exact model a couple of weeks ago. You have to take practically everything out in order to remove the display assembley. HD, optical, logic board and the inner frame. Which is pretty much all of it. You should get the service manual. Or use the online version at http://www.pbfixit.com/Guide/33.0.0.html
The hinges are glued into the screen assembley, which itself is almost entirely glued together. There are two screws along the side edges, but the rest is glue. A sharp, thin bladed knife is required to take the diplay apart any further. Run the blade carefully under the edges of the front bezel. The bezel will peel up happily enough, but you'll notice it tends to curl. Do not let this happen. You must flatten it again as you go, if it gets all curled, the metal will kink when you attempt to straighten it. And then it will look crap when you're done. This is the voice of experience talking, I had to get a replacement front bezel, as I murdered the first one. So take care and take your time.
You should be able to remove the broken hinge by running your knife around it and carefully levering it out from the display housing. To glue the new one in place, and to reglue the front bezel back on, I recommend a heavy duty two part epoxy. Over here in the UK, the most well know brand is probably Araldite. I don't know if they sell it elsewhere or not. Glue the hinge in, make sure it goes in nice and flat. Lying the whole thing on a flat surface will actually pull it away from the housing ever so slightly, so prop it up on each coner and weigh it down accordingly. Its a good idea to prepare this before you do any glueing, and double check that all will sit as it should. You're going to need to leave it to dry for a while. Overnight is best if you can stand it. I glued the hinges in first, then the front bezel back on the following day, but once the hinges are set in place you should be OK to glue the bezel back on.
Then leave it overnight, propped and weighted, and you can put the whole thing back together in the morning.
Sounds like fun don't it?
Its not that technically challanging actually, and well worth being brave since most Mac repairers will charge you at least 2 hours labour (more than you paid for the laptop), though even if you pay someone to do it, in your case, you could still sell it for a pretty good profit when its done.