It's a bit of a national myth, especially in America, that the Battle of New Orleans "won" the War of 1812 for the US. In reality, it didn't win anything at Ghent. It did instill confidence in the young nation that they held their own against British veterans and could defeat them.

The Treaty of Ghent ended the war in December 1814, with all sides agreeing that it was a draw. No borders changed and any territorial conquests were returned. Some of the most important grievances the US had claimed as motives for war -- naval impressment and neutrality rights -- weren't even addressed in the treaty. So the treaty was signed before New Orleans happened, though not yet ratified.

But the news took a month to reach American shores. During this time, Jackson had won at New Orleans in January 1815. Many Americans at the time believed (mistakenly) that, upon news of peace at Ghent, the battle somehow won the war or at least forced the peace. American statesmen eagerly perpetuated this narrative in the lead-up to the treaty's formal ratification in February.

British aims during the war from 1812 right up until 1814 were limited. They basically were concerned about the defence of their colonies in Canada, which was not a certainty with British resources overseas stretched thin due to the Napoleonic Wars. The war hawks in Congress, most from the frontier, agitated for war due to their desire to expand westward into territory in the Ohio Valley that was nominally indigenous land with a loose association to British interests. This land had long been promised by Britain to the first nations peoples since the end of the Seven Years War in 1763, but settlers kept encroaching on it. The British colony of Upper Canada was directly in the path, and most of the fighting in fact occurred on either side of the Niagara River. The bloodiest battle of the war, Lundy's Lane, occurred within sight of Niagara Falls.

1814 was a critical year as this was when Napoleon abdicated, freeing up British supplies and reinforcements to fight in America. This alone stopped the impressment of US sailors. With France defeated, there was no longer a demand for high volumes of sailors to keep the British navy afloat.

No longer concerned with defence, with fresh reinforcements from the Napoleonic campaigns in Europe, the British began aggressive punitive campaigns in 1814 along the Atlantic seaboard, in Chesapeake Bay, the Hudson Valley and New Orleans. The White House was torched, largely in retaliation for the US burning of the provincial capital at York (Toronto) in 1813. The British naval blockade strangled the US Atlantic trade and British veterans from Europe had captured Maine and even renamed it New Ireland.

By the end of 1814, Britain was weary of a quarter century of war and the US government was on the verge of bankruptcy due to the war effort. American negotiators in Ghent had to stall for time, as the British were pushing to keep any territorial conquests aka uti possidetis. Britain reneged on its promises to its native allies in the Ohio Valley, leaving them with nothing. Wellington had urged British diplomats to settle the peace quickly. Both sides ultimately agreed to status quo ante bellum aka the state of affairs returned to what it was before hostilities began.

While the Battle of New Orleans was a symbol of American confidence and military pride against its former colonial masters, it happened after the terms of the Treaty of Ghent were already settled and signed.

Sources:

Lohnes, Kate. "Battle of New Orleans". Article. britannica.com. 2018.

Wallenfeldt, Jeff. "Treaty of Ghent". Article. britannica.com. 2019 (updated).

Marsh, James H. "War of 1812". Article. thecanadianencyclopedia.com. 2012.