No, Panakeia didn't accompany her father to Eleusis. He brought his other daughter.

In case you didn't choose the Panakeia option for the last question, here's a recap: she was the goddess of cures and medicines, and she's mentioned, along with Asclepius himself and his other daughter, in the ancient Hippocratic oath. The prefix pan- means "all" in Greek, as we saw in the case of Pandora ("all gifts" or "giver of all"). In the case of Panakeia, she is the curer of all ills. It is from her that we get our English word "panacea," something that solves all problems or brings about a universal cure.