MacBook mid 2010 to body of MacBook air 2010?

Like for like is possible.

That is you get either a new case or find a dead unit which is the exact same model series. To find out what is the same review EveryMac like this:

All of the MacBook Airs: Full Listing. Then isolate out yours as example: Apple MacBook Air "Core 2 Duo" 2.13 13" (Mid-09), then isolate out what are the alike models which use the same cases: MacBookAir2,1. These are the units you can swap parts between.

Lastly, if you can find the part numbers of the parts you want to replace you can look them up on the net too (Bing/Goggle).

Now the bad news: You may find the damaged unit which you are going to use may not be what it looked like from the seller once you have it in your hands. So be careful on how you pay for it and make sure you can return it. Even still its rare to find & replace the parts as cheaply as just buying a good working system (new or used).

You may find just buying some snap on covers will make your system look good as new! with the least amount of costs!