How do I connect my 27inch 2012 iMac to my new MacBook air as an extra screen?

You do not, Target Display Mode is a dead technology and it will no longer work on newer Macs. Sorry, before you ask, no there are no work arounds. You should look over Target Display Mode (Target Display Mode) and you will see that I am correct. Your best option is to buy an external display for your MBA and connect it. This is not as expensive (does not require expensive cables, is much more reliable than Target Display Mode ever was, it simple to setup and external displays can be found easily for about $100.

Sorry to tell you what you did not want to hear, however it is correct.