The quick answer here is I would return it to the guy who sold it to you and get my money back ASAP.
If the connector on the logic board is damaged you may still have problems even with the cable removed as they could be shorting across (preventing you from booting up from an external source). You'll need to use a magnifier and make sure none are touching each other.
As Machead stated you could use an external case to hold your HD (or if your unit has a SSD) and connect it directly to the external USB port of your MBA. So you could at least use the system (Assuming the HD units connector is OK). What a pain to use in this state!
If you've got the case already and put the HD/SSD inside, can you even see the files on the HD/SSD when connected to another Mac? If you can you should be OK. If the OS is missing but you can access the HD/SSD you could try reinstalling the OS if you got the USB thumb drive with the OS on it that came with the MBA using your other Mac to prep the HD/SSD. If you need to use your the MBA to prep the drive you should be able to use a powered USB hub to connect both through the single USB port the MBA has.
If the HD/SSD is also bad it makes sense to bring it into Apple and get the logic board exchanged out and get a new HD or SSD. I'm sure you'll end up paying more than buying a new MBA doing this when you add in the cost of the MBA you paid the guy.