Is it obligatory to fulfill a vow to fast for a whole week after a child's recovery from birth defects, knowing that her treatment will continue for another year?

Fulfilling a vow of obedience is obligatory, based on the Prophet's (peace be upon him) saying: "Whoever vows to obey Allah, let him obey Him." As long as the child has not yet recovered, you are not obligated to fast the vowed week until a complete recovery has occurred.