I suggest you contact Asus directly. This desktop model has a 350W PSU and can come with a 6.5" GPU, such as HD8350. However, most 660s are around 9.5" That's much longer then the one you might have. Also, most systems with a 660 require a minimum of 430W. You might get by with 400W, but 350W seems too low.
Have you thought about removing the components and putting them into a bigger case with a larger PSU?
also if your pc is not neat inside, like wires and cables are everywhere, make sure to tidy that up. i had a hard time getting my 670 in my old mid tower case, it had about 2cm of room left in length and those 2cm were taken up by two large cables that attatch to the motherboard
That Asus case you have looks really nice. I hope you can fit a new PSU and GPU in there. If you are going to reuse the case make sure the new PSU has the same dimensions as the 350W one so it'll fit in the case.