Jackson started his fight against the British on Dec. 23, 1814, when 24 Americans and 46 British killed; the 'battle' was on January 8, 1815 at Chalmette Plantation. I don't remember when Jackson marched to New Orleans, he was in the City in December. The first verse of the song with date 1814 is correct as written.

Put Battle of New Orleans in the Search box and both DT and thread links come up.

That Lego simulation is cute, but the fort shown didn't exist. Earthworks only.