"Maybe not a rout but the battle was decisive. - kendall

Not really, by definition a "decisive" battle is one that decides an issue. Peace had already been agreed before the battle took place but neither side was aware of it.

The British Forces did not leave after the battle, New Orleans was only one of a number of targets, they decided to leave New Orleans even although they received reinforcements and a seige train to take the city after the battle. One month after the battle of New Orleans they took Biloxi in Mississippi then captured Fort Boyer in Mobile Baywere actively engaged in operations against Mobile itself, when word reached them that peace had been agreed. Immediately on hearing that the British withdrew in accordance with what had been agreed.

"Also, as an aside, Packenham was Wellington's Brother in Law and they didn't send Wellington, because Wellington was pro-American." - Bert

They didn't send Wellington because Wellington was far, far busier elsewhere acting a British plenipotentiary at the Congress of Vienna, I do not quite know why the song starts with "In 1814 we took a little trip" when the battle was actually fought in 1815.