Brokensha was born in Nailsworth, South Australia. He 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 Australian pianist Bryce Rohde; they formed the Australian Jazz Quartet/Quintet the following year with Australian bassoonist/saxophonist Errol Buddle and American saxophonist/flutist/bassist Dick Healey. This ensemble (sometimes recording as a quintet or sextet) toured together until 1958 after a tour of Australia.
Brokensha moved to Detroit, Michigan, 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".
In the 1970s he ran a steakhouse high up in a Downtown Building whilst working at Motown. The club was called "Brokensha's", and was located close to Detroit's Fisher Theater. It was a relatively small club with good food and, of course, Jack's great music. He was occasionally accompanied by his friend and fellow Detroit resident, pianist Bess Bonnier.
Following further tours in Australia 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.
- As leader
- And Then I Said (Savoy Records, 1963)
- Holiday Inventions (US Steel, 1968)
- Boutique (AEM Record Group AEM 25681-2, 1993)
- XMazz (AEM Record Group, AEM 25711-2, 1993)
- With the Australian Jazz Quartet
- "Australian Jazz Quartet" (Bethlehem BCP-1031, 1955)
- "The Australian Jazz Quartet" (Bethlehem BCP-6003, 1955)
- "Australian Jazz Quartet/Quintet" (Bethlehem BCP-6002, 1956)
- The Australian Jazz Quintet at the Varsity Drag (Bethlehem BCP-6012, 1956)
- Australian Jazz Quintet Plus One: Jazz in D Minor (Bethlehem BCP-6015, 1957)
- Rodgers & Hammerstein (Bethlehem BCP-6022, 1957)
- Free Style (Bethlehem BCP-6029, 1958)
- Three Penny Opera (Bethlehem BCP-6030, 1958)
- Reunion! Recorded Live - Adelaide Town Hall (AEM, AEM 25801-2, 1994)