Since John Eaton was just Jackson's friend and not an actual member of the administration, Jackson really didn't need to do much at all. Why touch this tawdry mess unless you're looking for trouble? Jackson getting involved would be like President Obama getting involved in Anthony Weiner's "Weinergate" scandal of 2011.
"You know what I found inappropriate? He didn't text me anything."
How He Handled the Scandal:
Almost completely out of spite, Jackson inserted himself directly into the middle of the controversy by very publicly siding with Eaton and nominating him to be the secretary of war. Foolproof, right?
Having lived through seeing his own wife slandered in a similarly public manner, the paranoid and defensive Jackson vigorously defended the Eatons against the accusations of the rest of his Cabinet. Eventually, Jackson became convinced that the whole thing had been the work of his political detractors to try to destroy his presidency (he was crazy, you see). The more the press raged against the Eatons, the more paranoid Jackson got. Not wanting to look weak by asking Eaton to resign from the Cabinet, but also not wanting to get eaten alive in the press anymore, Jackson finally asked for the resignation of the entire Cabinet, figuring that he could just start over again fresh after the next election.
"Maybe I'll fire the whole fucking country. I'll start over with a NEW America. Test me on this."
After the mass resignation, Samuel Ingham, the former secretary of the treasury, was pretty pissed at Eaton and sent him snarky letters. Eaton challenged Ingham to a duel, and when Ingham refused, Jackson offered Eaton the sensible and not at all insane advice, "If he won't fight, you must kill him."
Thus, on the advice of the president of the United States, Eaton got a bunch of his buddies together to hunt down the former treasury secretary, who managed to escape town with his own band of merry men. There was a presidentially sanctioned manhunt because of a scandal that was barely a scandal to begin with. Man, people were bored before television.