Arguably, Andrew Jackson might have already earned enough glory to be a famous man even if word of the peace treaty had reached New Orleans earlier, and the two armies not clashed on "the Glorious Eighth of January." In 1814, he had defeated the Red Stick Creek Indians at the Battle of Horseshoe Bend in Alabama, effectively breaking the power of the Creek Nation in that part of the country.
Still, nothing like cleaning the clock of a British army to win your laurels. Like most battles, it was a complicated business, but especially important in the victory was that the Americans were able to rain fire down on the British regulars, who were the victims of a logistical oversight: they didn't have the ladders they needed to scale the Americans' earthworks. Oops. History turns on small things sometimes.