How to copy data from non starting MBA?
I would started with getting that drive out of MBA with the instruction from iFixit for example.

Then I would buy one of these IDE to 40 pin ZIF adapters and find yourself old PC or Mac with IDE connection.

Fire up DiskWarrior or R-Studio (on Windows) or better and cheaper with DDRescue.
If you want to check if HDD is physically sound you might also try MHDD or Victoria tools (they are DOS and Windows only though).

And if you want to replace that HDD better go with SSD. At least 64GB SSD will do big boost to this old bugger. Provided you have Nvidia in yours. If it is intel I would just replace HDD and sell it.