Michael Cohen had many reasons to play ball last weekend when his legal team sat down to talk to federal prosecutors.

The Manhattan U.S. Attorney’s office had testimony from Mr. Cohen’s accountant and business partners, along with bank records, tax filings and loan applications that implicated not only Mr. Cohen in potential criminal activity, but also his wife, who filed taxes jointly with her husband. Prosecutors signaled Mr. Cohen would face nearly 20 criminal counts, potentially carrying a lengthy prison sentence and staggering...