From what I understand, you are responsible for paying all employee taxes for a home health aide. This includes payroll taxes, social security taxes, unemployment taxes... the works! Google around for the "nanny tax" to get the particulars.
This is a major reason to go through an agency. If you are paying the agency and not the aide, then you aren't the aide's employer and you probably don't have to worry about the taxes angle.
I'm just some guy on the internet, not a tax attorney, so make sure to get some professional advice.