How did Holmes convince Laura Lyons to confess?

Mr. Holmes revealed that Stapleton, the man who had duped Lyons, was actually a married man. She no longer had any reason to protect him..... he'd been false with her.

“His wife!” she said again. “His wife! He is not a married man.”

Sherlock Holmes shrugged his shoulders.

“Prove it to me! Prove it to me! And if you can do so—!”

The fierce flash of her eyes said more than any words.

“I have come prepared to do so,” said Holmes, drawing several papers from his pocket. “Here is a photograph of the couple taken in York four years ago. It is indorsed ‘Mr. and Mrs. Vandeleur,’ but you will have no difficulty in recognizing him, and her also, if you know her by sight. Here are three written descriptions by trustworthy witnesses of Mr. and Mrs. Vandeleur, who at that time kept St. Oliver’s private school. Read them and see if you can doubt the identity of these people.”

She glanced at them, and then looked up at us with the set, rigid face of a desperate woman.

“Mr. Holmes,” she said, “this man had offered me marriage on condition that I could get a divorce from my husband. He has lied to me, the villain, in every conceivable way. Not one word of truth has he ever told me. And why—why? I imagined that all was for my own sake. But now I see that I was never anything but a tool in his hands. Why should I preserve faith with him who never kept any with me? Why should I try to shield him from the consequences of his own wicked acts? Ask me what you like, and there is nothing which I shall hold back. One thing I swear to you, and that is that when I wrote the letter I never dreamed of any harm to the old gentleman, who had been my kindest friend.”