Hi there,
It's not possible to transfer HIV via hands, as the virus becomes inactive when exposed to air. It's also not possible for blood to go from the nurse's hands to the injected material because of the way the equipment (syringe and vials) is made. Please rest easy -- there's no way your daughter could have been exposed to HIV, and giving her PEP is unnecessary. PEP can also have a lot of very unpleasant side effects, so it's generally given only in cases of known exposure.
If you're still concerned, there's no need to wait 6 months to test again -- you can do it when you return for your daughter's last vaccine shot. Hope this helps, and hope your daughter's recovering from that bite!