Hi, and thanks for your question.
We really don’t see HIV transmitted this way. HIV does not live outside of the body on surfaces or objects. Even if there was HIV-infected blood on the cotton ball, the virus would die within a few seconds and would not be able to infect an individual. From what you’ve described, there is no risk of HIV to your daughter. If you are concerned you could always visit your family doctor or pediatrician for further information and guidance.
Let us know if this does not answer your question or if you have any more questions or concerns.