After schooling at St Aloysius College, Adelaide, where she was dux in 1929 and 1930 Mitchell studied law at the University of Adelaide. Whilst an undergraduate on finding that, as a woman, she was barred from joining the Law Students' Society Mitchell led the push for the establishment of a female equivalent. Her degree was conferred in 1934 and she won the David Murray Scholarship, awarded to the most outstanding student of the year. In December of that same year, at the age of 21, Mitchell was admitted to the Bar. In practice she specialised in matrimonial and industrial law and began lecturing in family law at the University of Adelaide.

Mitchell agitated for women to be allowed to sit on juries; a reform which was legislated in late 1965. In 1962 she was appointed Australia's first female Queen's Counsel and in September 1965 Australia's first female judge of a superior court (Supreme Court of South Australia). Mitchell advocated for equal pay for equal work for women and was Australia's representative to the United Nations conference on the Status of Women in Family Law in 1962. She was Chair of the Criminal Law and Penal Methods Reform Committee of South Australia (1971-1977) which instigated reforms in the areas of court procedures in cases of rape and other sexual crimes, protection of the rights of the accused, compensation for victims of crime and the bail system. Later she was also Chair of the South Australian Parole Board.

In 1981 Mitchell was appointed by the Fraser government to the Chair of the newly established Commonwealth Human Rights Commission, becoming a passionate advocate for all disadvantaged people in Australian society. Mitchell's belief in assisting the disadvantaged continued later in a most practical way when she spent several years delivering meals to the elderly by volunteering with Meals-on-Wheels. In 1983 she was elected to the chancellorship of the University of Adelaide; the first female chancellor of an Australian university and the first woman chancellor, apart from royalty, of any university in the British Commonwealth.

In 1991 Mitchell was made Companion of the Order of Australia (AC) and became Governor of South Australia. She was the first female governor of an Australian state and held in high regard by politicians and the public throughout her five year appointment.

A statue of Mitchell stands outside Government House on North Terrace, Adelaide, and, in recognition of her lifelong love and support of the arts, the Adelaide TAFE performing arts centre on Light Square Adelaide, was named in her honour.

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December 1934: Admitted to the Bar

1962: Appointed Queen's Counsel; first Australian woman Queen's Counsel

23 September 1965-October 1983: Judge of the Supreme Court of South Australia; first woman to be appointed to a superior court in Australia

1971-1977: Chair of the Criminal Law and Penal Methods Reform Committee of South Australia

12 June 1971: Made Commander of the Order of the British Empire (CBE)

1974-October 1981: Chair of the South Australian Parole Board

July 1978-September 1981: Chair of State Heritage Committee of South Australia

10 December 1981-9 December 1986: Founding chair of the Commonwealth Human Rights Commission

1981-1990: Member of the Council for the Order of Australia

12 June 1982: Made Dame Commander of the Order of the British Empire (DBE)

1983-1990: Chancellor of the University of Adelaide; first woman Chancellor of an Australian university

26 January 1991: Made Companion of the Order of Australia (AC)

1991-1996: Governor of South Australia; first woman Governor of an Australian State

1996-2000: Chair of the Ministerial Board on Ageing (South Australia)

1 January 2000: Made Commander of the Royal Victorian Order (CVO)