Continuity mistake: Strange is at home, looking for help on a video chat. His friend says no, and in his rage he sweeps almost everything off the table including the Microsoft Surface laptop. Christine enters with food, we cut back, and the area of the table to his left, which was totally clear a second ago, now has several pieces of paper on it and the laptop with a closed lid. A few moments and cuts later, the laptop is in perfect condition on the table with lid open.
Continuity mistake: When Hong Kong is being destroyed, a vendor's sign falls to the ground, written in English. However, in the shot where the city is being restored, the same sign is now in Chinese.
Continuity mistake: When Kaecilius is attached to a wall just his hand is free, but in the next shot now his head and other hand are free too.
Continuity mistake: The wound and blood on Strange's left eyebrow is first fresh, then dry, then it's gone, then is back again.
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.