Pharaoh's daughter did not accidentally find Moses. According to Sotah 12b, she came to the river to immerse herself—not for bathing, but to wash away the spiritual impurity of her father's idolatry. Rabbi Yohanan said, in the name of Rabbi Shimon ben Yohai: she was converting to the God of Israel. The word "to bathe" (lirhotz) carries the same connotation as spiritual cleansing in (Isaiah 4:4): "When the Lord shall have washed away the filth of the daughters of Zion."
Her handmaidens tried to stop her from rescuing the baby. "Our mistress, when a king decrees a decree, even if the whole world disobeys, his own household must obey. Yet you defy your father?" The angel Gabriel struck the handmaidens dead—all but one, because a princess should not stand alone.
The Torah says she sent her amah to retrieve the basket (Exodus 2:5). Rabbi Yehuda and Rabbi Nehemya disagreed on the meaning. One said amah means her arm—her physical forearm. One said it means her maidservant (ama). If it means her arm, the Talmud says something extraordinary happened: her arm miraculously stretched across the distance to reach the basket. The Hebrew amah also means "cubit"—her arm extended many cubits beyond its natural length.
When she opened the basket, she saw the baby crying, and she had compassion. Rabbi Yosei bar Rabbi Hanina said she saw the Shekhinah (שכינה), the Divine Presence, with him. The child was glowing.
Moses refused to nurse from Egyptian women. Rabbi Elazar explained: a mouth that would one day speak with God directly could not suckle from impure sources. So Miriam, hovering nearby, stepped forward and offered to find a Hebrew wet nurse—Moses's own mother, Yocheved. Pharaoh's daughter unknowingly hired the baby's biological mother to raise him, and paid her for it.