There were no Canadian units present at the burning of Washington. Moreover while we can debate if the United States or Britain won the war, Canada definitely did not. While the war ranks third in the public memory of historical events for Canada and was a driving force in Canadian nationalism the fact remains that Canada was not a political entity. In the same sense that the United States didn't defeat France in the seven years war, Canada didn't "win" the War of 1812. Moreover Britain had some 45,000 regulars in the North American theater by war's end, a force that far outstripped the number of Canadian irregulars. These regulars also provided the overwhelming bulk of forces used to invade the United States in 1814. Britain also provided the naval forces used to raid the United States and cripple the American economy by summer of 1814. While Canadian forces were important in the early war battles, Britain provided the bulk of the fighting men and war ships.

The British burned down the White House in 1812, but they didn't really win the war