Complex sentence: He then went into partnership with Jack Harris (an American bandleader), and in 1937 they bought a club together (Ciro's Club); they alternated performances there until a disagreement led to the rupture of their partnership.
Split into simpler sentences: He then went into partnership with Jack Harris (an American bandleader), and in 1937 they bought a club together (Ciro's Club). Ambrose and Harris alternated performances in Ciro's until a disagreement led to the rupture of their partnership.