Wellington was just about to be posted to America. He was due to follow his Peninsula Army heading there. A lot of the regiments that gained distinction against France were sent to deal with the upstarts across the sea. Regiments such as the 60th KGL were sent over, with attached officers from many regiments, including the 95th Rifles, one of the most highly decrated units in the army and with some of the most battle honours.
While it is true that the Americian navy triumphed on several small engagements against the Royal Navy, when it came down the last final full scale battle, they were beaten. One the other side, the British navy was successful in almost every engagement until the final battle of New Orleans. Hence the new units sent out. Had the United States not ended the war, they would have been crushed.
As others have said, the War of 1812 was a side show to the Peninsular War. When Napoleon returned in 1815, the units in America were immediatly sent back to Europe. Some men had basically just arrrived in the US when ordered to return home at full speed. Officers such as Harry Smith were turned around and sent back to deal with the 'real' threat.
As for the White House, it was men of the 44th Regiment of Foot who hold the crown for burning it down. They still haven't forgotten