Brokensha was born in Nailsworth, Adelaide, Australia. He initially studied percussion under his father, and played xylophone in vaudeville shows and on radio. He played with the Australian Symphony Orchestra during 1942-44, and played in a band in the Air Force in 1944-46. Forming his own group, he played in Melbourne in 1947-48, Sydney in 1949-50, Brisbane later in 1950, and Adelaide in 1951. In 1953 he moved to Windsor, Ontario, Canada with pianist Bryce Rohde; they formed the Australian Jazz Quartet the following year with Errol Buddle and Dick Healey. This ensemble (sometimes recording as a quintet or sextet) toured together until 1958 after a tour of Australia.
Brokensha then moved to Detroit, Michigan, USA, where he was hired by Berry Gordy of Motown Records as a percussionist, becoming one of the few white members of Motown's Hitsville U.S.A. recording studio's house band, The Funk Brothers. He was given the nickname "White Jack", to distinguish him from Jack Ashford, an African American percussionist nicknamed "Black Jack". Following further tours Down Under with Sammy Davis, Jr. and Stan Freberg, Brokensha founded his own music production company. He did a session with Art Mardigan in 1963, and after this became more active in disc jockeying and writing music for television. He recorded as a leader again in 1980 and continued to lead his own group well into the 1990s. The Australian Jazz Quartet also reunited for tours and recording in 1994.
Jack died in Sarasota, Florida, of complications from congestive heart failure at the age of 84.