In a nutshell, it looks like your problem is that you have a slim case, and all slim cases require low profile add-on cards. Your options are:
A) Cut a hole in the side of the case so the graphics card has room to stick out. You would need to figure out a way to secure it because you wouldn't be able to screw the bracket into the case and wouldn't want it to short out or the fan to get damaged by it coming into contact with anything.
B) Buy a new, full sized (Mid Tower or something along those lines) case as others have mentioned and move all your hardware over. You have a specialized power supply, so you might need a new one of those as well, but all told you should be able to do it for around $50.
C) Exchange the card you bought for a low Profile card like the
PowerColor AX7750 1GBD5-HL. That card will probably be among the best performers that will fit your case. Unfortunately a quick search of Newegg's product list didn't come up with any low profile 7770's. The additional nice thing about that specific model is that is comes with a second low profile bracket for the VGA port. So you would be able to have all three connections available if you are OK covering two slots. My Newegg customer service experiences have always been superb, so even if they say there is a 15% restocking fee you might be able to call them up and (if your are polite) get them to exchange the cards for you by explaining your situation/mistake. Even if you have to pay a restocking fee this will still be cheaper than going the new case route.