Must I pay half the price of the calf that my mother-in-law vowed, without my knowledge, for my daughter's recovery?

There is no vow for the son of Adam concerning what he does not own. Therefore, your mother-in-law's vow is binding regarding the half calf that she vowed on her own behalf, provided that she intended it as charity and an act of worship. As for what she vowed on your behalf, it is not binding, and you are not obligated to pay her anything. She must fulfill her vow by participating in a sacrifice to the extent of the half she vowed. This vow, which is contingent upon a disliked condition, is [problematic]. However, if you were to pay your mother-in-law the value of half of that sacrifice as a gesture of gratitude to Allah, kindness towards her, and to relieve her of embarrassment, that would be good and an act of charity on your part.