You only need to issue 1099-Misc to people that you used in connection with your trade or business.
If you hired the teacher aide outside of your business, then you do not have to give her a 1099-Misc. It is up to her to report this income. Her expense is obviously not deductible in your trade or business, but perhaps you can claim it elsewhere depending on the circumstances.
If you do issue a 1099-Misc to the aide, you cannot include it on your business 1096 because you are only allowed to put expenses related to your trade or business. together.