Business in Florida, doing billing in Georgia
Assuming this individual??s job duties are going to stay basically unchanged, you will need to register as a Georgia employer and withhold Georgia income tax and pay Georgia unemployment tax on her wages. You will need to register with both the Georgia Department of Labor and the Georgia Department of Revenue. If you are using a payroll service they may be able to assist you with the registration.
If the individual??s job is changing enough that they now qualify to be an independent contractor, you would not need to register with Georgia, but would instead pay this person as a vendor and likely need to send them a 1099 instead of W2 at year end. The rules for who qualifies as an independent contractor can be found here.
Finally, unless you are going to be selling in Georgia, which I doubt a medical practice will do, you don't need to register the company as a foreign entity doing business in Georgia because it will not have any Georgia revenue.