All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If this person who intends to perform Hajj, and it is due upon him to perform it, is responsible for looking after that patient and there is no one else to do this task, then there will be no sin on his part to postpone the Hajj of this year because it is permissible to postpone Hajj for a legal excuse. In fact, looking after this patient, especially when there is no one else to take care of him, is one of the apparent needs that make it permissible to postpone Hajj, Allaah willing.
However, if this patient is in no need for this person's care or there is someone else to take care of him, then it would be meaningless to postpone the Hajj. On the contrary, it is due on him to hasten to set out for Hajj once it is possible for him to do so because Hajj is obligatory immediately (on the one who is financially and physically capable of doing it) according to the preponderant opinion, which is also the opinion of the majority of scholars. He has to strive to supplicate Allaah to a good effect for this patient in the blessed sites of the Hajj; perhaps this may be a cause of his cure.
Allaah Knows best.