Kaecilius: How long have you been at Kamar-Taj, Mister..?
Dr. Stephen Strange: Doctor.
Kaecilius: Mister Doctor?
Dr. Stephen Strange: It's Strange.
Kaecilius: Maybe. Who am I to judge?
The Ancient One: You think you know how the world works? You think that this material universe is all there is? What is real? What mysteries lie beyond the reach of your senses? At the root of existence, mind and matter meet. Thoughts form reality. This universe is only one of an infinite number. Worlds without end; some benevolent and life-giving, others filled with malice and hunger. Dark places where powers older than time lie, ravenous and waiting. Who are you in this vast multiverse, Mr. Strange?
Dr. Stephen Strange: This doesn't make any sense.
The Ancient One: Not everything does. Not everything has to.
The Ancient One: We never lose our demons, we only learn to live above them.
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.