Is it permissible for a patient in a hospital to pray with the intention of facing the Qibla without actually facing it, if the direction of his bed is towards the Qibla and it is difficult for him to turn around, and he sometimes prays sitting and sometimes lying down?

We ask Allah to grant your brother healing and well-being. Your brother is obligated to fear Allah to the best of his ability in his prayer, so he should pray according to his condition: standing, if unable then sitting, if unable then on his side. If he is unable to face the Qibla (definition), he should pray facing another direction. Whatever conditions of prayer he is unable to fulfill are waived for him. He should try to acquire a bed that allows him to face the Qibla. If this is not possible and his request is not met, he should pray according to his ability, and there is no blame upon him.