Other mistake: When Dr. Strange rushes a patient to brain surgery they show him and Dr. Christine Palmer operating without surgical masks. When they cut away and come back they're wearing masks. Also, Dr. Palmer is an ER doctor, and her assisting him would never happen in real life.
Other mistake: After the fight where his sling ring is stolen by Kaecilius, he has a sling ring in his possession and uses it to jump to Hong Kong.
Answer: Medically, he could not heal his hands. He spent his entire fortune trying to return to being a surgeon through experimental procedures. Once he completed his training to become a sorcerer, he realised there was more to the world than he realised. The Ancient One correctly stated that he was a surgeon for his own ego; he wanted to prove he was the best and helped people for that reason only. By joining the sorcerers he was given a greater purpose and chose this over returning to his former life.
His hands were healing, just not quickly enough for his needs. This was showing his egotistical attitudes.