Factual error: In the first scene where Dr Strange is preparing for surgery he uses improper surgical gowning technique breaking aseptic protocols. Strange puts on a mask after washing his hands, contaminating his clean hands by bringing them in proximity to the non-sterile environment of his face. Surgical masks must be worn before the full washing of hands. Another error is that he inserted his hands all the way through his gown to don gloves. Hands must never leave the sleeves and gloves must be put on with the sleeves still covering.
Factual error: When Dr Strange first exits the London Sanctum there is a blue USPS letter drop box on the corner. London Postal boxes are a different design and red.
Factual error: When Tilda Swinton dies and is lying on the operating table, her pupils are not dilated as they would be immediately after death.
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.