The best thing here is to replace it. The cover also protects the antennas and wires which interconnect the base to the lid.
Your model has a different design than the newer models. The cover also supports the WiFi & Bluetooth antennas so its a bit more expensive and a bit more work to install.
You'll need to flow this guide to take the lid off: . And this is the part you'll need: [http://www.powerbookmedic.com/MacBook-A...MacBook Air 13" Mid 2012 Display Assembly Replacement]. Follow the guide to the end at which point you can pop off the cover and replace it, then reassemble.