I dropped off my daughter at nursery school today.
I took my daughter to her preschool this morning.
この場合、次のように言うと自然で良いと思います。
ーI dropped off my daughter at nursery school today.
「娘は今日保育園に預けました」
to drop off someone at nursery で「人を保育園に預ける」
ーI took my daughter to her preschool this morning.
「今朝、娘を幼稚園に預けました」
to take someone to one's preschool で「保育園に預ける・連れて行く」
ご参考まで!