Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
No doubt a doctor is responsible for a mistake that causes harm to his patient. The responsibility increases or decreases depending on the extent and the type of his mistake. Islamic Shariah takes into consideration the medical responsibility as the Prophet (Sallallahu Alaihi wa Sallam) said, reported by Amr Bin Shuaib from his grand father that: "Whoever gives medical treatment without having the knowledge of the profession is responsible (for the harm he caused). (Reported by Ibn Majah and Al-Hakim who considered it to be sound and Imam Al-Zahabi agreed with him.
A poorly qualified doctor commits a sin when he engages in treating people and is also responsible for his mistakes.
However, in the unintentional mistakes that happen after a qualified doctor has taken all necessary precautions, the doctor does not commit a sin for such mistakes, but he is responsible for the damage or loss he has caused. For example when a surgeon's hand slips at the time of surgery, or something like that happens by X-Ray technician, anesthesiologist, etc, and it causes physical harm to the patient, in such an incident one does not commit sin but he has to pay compensation for the damage caused. If a doctor is experienced, well-qualified, and if he tried his best to serve the patient, but despite his efforts the patient unintentionally is harmed, the doctor is not responsible for that since he did not make any mistake.
In the light of the above details if the doctor made a mistake that caused your wife harm, or your child became handicapped then you have every right to demand compensation. If you think doing so will be difficult and time consuming then be patient, or forgive the doctor, Allah will reward you with the best. Allah says: "And to forego is nearer to At-Taqwa (piety, righteousness, etc.)". (Al-Baqarah 2:237)
Desire of taking revenge is from Satan, so do not heed to it.
Allah knows best.