I think Apple's maximum flat repair charge for most laptops is in the $700 to $750 range, and I have a feeling that's what they'll charge you.
That said, that might actually be equal to or cheaper than the cost to repair it yourself. The display is a
single part including external shell that runs for $599 ($499 if you're willing to accept blemishes, which I have a suspicion you won't), and the upper casing (the part where your wrists rest when you're typing, and the sides of the case around the keyboard) is
another $299 new, $249 used. If there's damage to the lower/bottom case (the hatch at the bottom of the MBA),
that's another $99, and is a used part.
Of course if you go this route it's on you to try to do all the labor, which might get you into some trouble if a screwdriver happens to slip. So, while USUALLY Apple repairs are more expensive, this time it might actually be competitively priced.
Also, do you have renter's insurance or homeowner's insurance? Check with them, they might be able to help you out with the cost of repairs.
[EDIT: Realized that the upper case is in fact, damaged on the OP's Mac, while the bottom case might not be.]