Must I fast for another sixty days after I vowed to fast for sixty days when I hit my daughter, then hit her again in a moment of anger?

The vow you made is called a vow of obstinacy (Nathr al-Lajaj), and it does not require fulfillment of the vow. Rather, it requires an expiation for breaking an oath (kaffarat yameen) if the condition upon which it was made occurs, which is the striking. A vow of obstinacy (Nathr al-Lajaj) refers to persistence in contention, and what is meant by it is that which takes the form of an oath, where the one making the vow intends to prevent himself or others from something, or to encourage something, by committing to an act of worship. In this case, if the condition occurs, an expiation for breaking an oath is required, based on the saying of the Prophet, peace and blessings be upon him: "The expiation for a vow is the expiation for an oath."