Is the fast of one who did not intend to fast on the night preceding the last days of her menstrual period valid, thinking that she had not yet purified herself, then realized during the day that she had become pure, and performed ghusl after the Asr prayer?

A ghusl (ritual bath) from major ritual impurity (janabah) or menstruation is not a condition for the validity of fasting. However, your fast on that day was not valid for two reasons:

First: Your seeing the white discharge (al-Qassah al-Bayda') during the daytime. One cannot be certain that purity occurred from the beginning of the day. A condition for the validity of fasting is that no part of the day passes while the woman is menstruating.

Second: Not having the intention (niyyah) from the night before, which is a condition for the validity of obligatory fasting according to the majority of scholars.

Therefore, you must make up that day's fast. The fasts of the days that followed are valid. You did well by refraining from eating for the remainder of that day, and its recommendation is more preponderant.