How to downgrade the OS to the original one on a MBA 2010?
The MacBook Air came pre-installed with Mac OS X 10.6 Snow Leopard, which was available with a physical installation media only.

Here is what I would do in your situation:

Parts Replacement (USB Key Installation Media):
Contact Apple and ask if they still replace the missing parts (costs will apply, but I do not know how much).



Order a Replacement Disc:
Ask, if they still sell replacement discs.  The costs were reasonable the last time I heard (~$15-20).
NOTE: The MBA being a Late 2010 may require an installation media unique to its build.  So a general-purpose replacement disc of Mac OS X 10.6 Snow Leopard may not work with it.  Ask Apple specifically if it works with your daughter's model before you decide to buy it.

Create a USB Thumb Drive from a Replacement Disc:
Should the general-purpose disc work, knowing that the MBA does not have an optical drive, if you have a different Mac with an optical drive, you can create a USB drive from it by just cloning it over using Disk Utility.

Buying an Optical Drive to Use Disc:
If creating the USB thumb drive is not an option for you, you may want to look into buying an optical drive and use that with the disc (obviously only if the general-purpose disc is an option in the first place).  The optical drive can be used for other things as well and even used with a potential new Mac computer lacking an optical drive.  So, the costs of $79 for the drive would not be wasted for a single purpose.