Transplanting MacBook Air 11" logic board to a 13" one
I'm not going to say it'd be absolutely 100% impossible, but unless you had time (and money) to burn on a side project, it really wouldn't be a practical exercise. By this I mean I can't see a way for you to then be able to use it in the usual manner one uses a laptop. 

For example, your USB and headphone/microphone ports won't line up properly, so how easy will it be to connect devices to them? Sure, you may be able to connect extension leads from the logic board connectors to the exterior of the casing, but this isn't practical on a stationery desk let alone a portable laptop.

The other issue is the display module itself. You don't specify the exact models of your MacBook Airs, but the display module needs to be able to connect to the logic board. In order to do so the cabling needs to be in the right location (or long enough) in order to reach the logic board. And, this assumes the connection is the same. Trust me, the cabling inside these devices are no longer than they need to be - everything is a very tight fit!

Some observations

Here are some additional comments:


If you did want to do this as a side project, you'd have to be willing to largely accept the device having to either sit stationery on a desk in order to use it, or be willing to do a bit of soldering and custom cabling in order to try and get it to work as a portable laptop.
Assuming you are willing to do the above, before proceeding you would have to check the cabling (type, location and length) of the 13" MacBook Air display module and confirm you can connect it to the 11" logic board. If so, then that's probably the hardest obstacle overcome (as I can't see a way for you to connect the display module otherwise).
If you have home contents insurance, you may find that your 11" MacBook Air is actually covered by the insurance policy. It would be easier to determine this and just make an insurance claim. You'll likely find that they deem it uneconomical to repair and will replace it for you.
If you don't have home contents insurance (or your 11" model isn't covered by it), since doing what you propose almost certainly restricts it to being a desktop Mac, have you considered just purchasing a used display from a garage sale (or similar) and using it with that? 
As for the 13" MacBook Air, I sense you may be approaching this in the classical XY problem manner. That is, you have a problem (X) and decide the solution is Y, so you've come here asking a question about Y rather than the actual problem X. To clarify what I mean, your 13" MBA doesn't boot and beeps at you, so you assume it's a faulty logic board. However, I suggest you ask a separate and specific question about that.