Not to get side tracked - but if your daughter was bleeding, it would require the person attending her to be bleeding or having an open wound in order for a possibility of transfer of the HIV virus. The mere handling of blood can not transfer the virus.

As to testing an employee - even if a test was positive you may not, not employ the person, that would be discrimination.
You could of course have parameters of care, failing which the employee is negligient. However you cannot say that when ever you touch my child, or when my child is bleeding you must wear gloves. That is discrimination, impugnes dignity and a host of others. A condition that when a person has an open wound they need to take precaution, would be reasonable.