The internal affairs of the Hospital and the making and enforcement of rules and regulations for the government and discipline thereof shall be vested in the board of trustees of fifteen members, which board shall be constituted as follows:
(a) The Mayor or his designee shall be an ex officio member of the board of trustees.
(b) One physician, a resident of and practicing in the City, to be elected upon the nomination of the organized professional staff of the Hospital.
(c) One lawyer, a resident of and practicing in the City.
(d) One woman, a resident in and a qualified voter of the City.
(e) Eleven other persons, residents in and qualified voters of the City.
The board of trustees constituted as set forth in this section shall be self-perpetuating, but Council shall confirm or reject all persons nominated thereto by the Board, as terms expire or vacancies occur.