In 2010, Judge Wood presided over the guilty pleas of 10 Russian spies who had worked as long-term sleeper agents — a case that inspired the TV show “The Americans.” Six years later, when she sentenced Dean G. Skelos, the former Republican leader of New York’s State Senate, to five years in prison on corruption charges, she admonished him for ignoring his “moral compass.”

Image Judge Wood Credit... Amy Sussman/Getty Images

“Through your crimes,” she said, “you have caused immeasurable damage to New Yorkers’ confidence in the integrity of their government.”

(Mr. Skelos’s conviction was later overturned on appeal after the Supreme Court narrowed the legal definition of corruption. He is scheduled to be retried in June, again before Judge Wood.)

The case of Mr. Cohen more or less landed in her lap. In a bit of legal happenstance, Judge Wood was the emergency duty judge last week when Mr. Cohen’s lawyers filed a motion challenging the search warrants that federal agents used to seize his cellphones, papers and computers in a series of extraordinary raids this month. It remains unclear if Judge Wood will continue to preside as the matter moves forward. If criminal charges are ultimately brought, the case would be randomly reassigned.

But at least so far, Judge Wood has shown herself to be a firm jurist in the proceeding. On Monday, she compelled Mr. Cohen’s lawyers to reveal that Mr. Hannity had once been one of Mr. Cohen’s clients. Engaging her in a game of cat-and-mouse, the lawyers had at first described Mr. Hannity merely as a “prominent individual” who was embarrassed to be connected to the Cohen investigation.

But after another lawyer — representing news media outlets — stepped in to argue that embarrassment was not sufficient grounds to keep the name a secret, Judge Wood decided she agreed. When Mr. Cohen’s lawyers pushed back, Judge Wood was blunt.

“I’m directing you to disclose the name now,” she said.

And so the lawyer did.

Although the moment passed with a burst of audible gasps, Judge Wood’s measured approach — not raising her voice, but not backing down — showed a side of her that lawyers in her court know well.