How do I replace broken left screen hinge

You are going to want to follow, click here, this guide to remove the display (upper clam shell). Then you are going to want to follow, click here, this guide from step 15 on for the actual hinge replacement. You should be able to find your part from one of these places, click here. A left hinge from mid 2009 to present should work. There is not much documented on Apple laptops for the 2011 models. For the 13" MBP, I know the 2009 & 2010 display assemblies, LCDs, hinges & LVDS/video cable interchange. I know the late 2008 through (and including) early 2011 15" MBP display assemblies, LCDs, hinges and video cables interchange. I see no reason why Apple would change the trend on this.

Your other option is to take it to an Apple Care provider and pay their price.