One of the most trusted allies of J.P. Morgan Chase & Co. Chief Executive James Dimon left the nation's largest bank to run a payment-processing company, the latest executive to depart after a multibillion-dollar trading blunder.

The 53-year-old Frank Bisignano was barely visible outside J.P. Morgan, but was well known within the company as a fixer of operational problems and was a close aide of Mr. Dimon since 2005.

Mr....