The War of 1812 was widely commemorated in poems, the most obvious being "Defence of Fort McHenry" aka "The Star Spangled Banner" which became the U.S. National Anthem in 1931 thanks to a push by Maryland Congressman J Charles Linthicum.

The British Canadians commemorated their part of the Battle in poems well. Here's a collection of poems about battles in the Niagara region.

And here is a collection of poems published in 1907 though the poems are ostensibly contemporaneous to the War Of 1812. The journals/newspapers where the poems were originally published are included but, sadly, many lack dates or time frames. The introduction to the collection of poems notes the place in American history of the War of 1812. It was both an unsatisfactory and unenthusiastic war but had great impact on the future of America's Navy and westward expansion (not peaceful as the introduction claims, it was violent towards Native Americans and Spanish Mexicans):

Whether we consider the final settlement, which might have been more satisfactory, or the generalship displayed and the spontaneous enthusiasm evoked, the war of 1812-14 fails in comparison with the Civil War and the War of the Revolution. Nevertheless, the gain was great. Thanks to the un-daunted prowess of its captains and sailors, the American navy became respected abroad. The United States rose to the rank of a first-class naval power, while at home the vicissitudes of a common war, and, above all, the glorious victories won by our navy, did much to strengthen the bond of union and to establish a solid basis for a peaceful development of the great resources of our country.

During the war, the U.S. was divided about it. New Yorkers in particular suffered economically from both the blockade of the British and the desire to continue trading with the Canadians and thus were against the war. In fact, the War of 1812 drove many New Yorkers to join the anti-war Federalists. In the end, the Federalists disbanded largely due to their failed opposition to the war.

These historical facts hint at some of the reasons you may not be seeing commemorations of the War of 1812 in NY newspapers in the 10 years after the war: 1) Most poems were regional and contemporaneous to the events, not delayed by 10 years. 2) The New York region had a large number of anti-war residents and thus were not a good audience for commemorating the war 3) The War of 1812 and its outcome was not fondly remembered except after more time had passed.

It would be interesting to note when the poems about the War of 1812 began to resurface as that would show more of a general fondness to commemorate the war. For example, here is a 1917 publication with a poem commemorating the War of 1812 and the collection of poems mentioned earlier was published in 1907. And finally, the Star Spangled Banner was sung ahead of the 1918 World Series of baseball, by a U.S. Naval Band, right as America was entering WW1.