There were certainly those in Britain who had businesses in Canada and the western territories who had hoped that the English negotiators in Ghent would manage to extract boundary concessions or even more territory for what's now western Canada, as it was obvious the Americans were expanding. And there was a strong wish for the American government to be punished for their impertinence. But there had already been a very long and protracted Napoleonic War, and at the same time as negotiations at Ghent with the US, there were very, very important negotiations at the Congress of Vienna that came from Napoleon's defeat. The Congress of Vienna was seen as far more important, and it was- it would, in the end, indeed restrain European wars for decades- and it seems to have been pretty commonly accepted that, like it or not, the war with the annoying Americans had to be quickly settled in order to have a stronger hand in Vienna.
There was also simply the question of what could be done to the US. It was one thing to land and burn the capitol; that only required a modest-sized professional force. But to actually manage to seize and occupy the large American territory would have required many, many times that. The British had already found it difficult to chase down and defeat American forces in the War for Independence, when there were only 13 colonies. By 1814, the US had greatly expanded, and a campaign in a vast wilderness would have been even harder. And, in 1814, Britain's military was more stretched and badly worn than in 1776. The British delegation tried for the establishment of a Native nation as a buffer in the Great Lakes region, between Canada and the US- something that had actually been British a goal since the end of the War of Independence- but they gave it up.
Perkins, Bradford. (1964) Castlereagh and Adams : England and the United States, 1812-1823. Univ. of California Press
Between 1803-1805, Napoleon had massed 200,000 troops for a potential invasion of Britain. A flotilla of troop transports were gathered across the Channel. While the Battle of Trafalgar in 1805 ended the imminent danger, this would remain an existential threat as Napoleon conquered the rest of Europe.
Britain was in a total war against France -- it regarded the War of 1812 as an inconvenient distraction, a sideshow to Britain's war for survival against Napoleon. Britain did not want to spend more imperial resources to defend Canada than they had to, hence the largely defensive strategy they pursued there until 1814.
In 1814, once Napoleon was forced to abdicate, there were rumblings in Britain that the Duke of Wellington should lead British forces in their punitive campaigns in America. While he didn't object to being sent overseas if duty called for it, he made an astute observation that the only path to success in North America would be to control the waterways, specifically the Great Lakes basin -- a hotly disputed area where Britain and the U.S. were in a frantic ship-building arms race, with no end in sight.
News of the 1813 American naval victory on Lake Erie would have arrived by then. The entire Niagara frontier on both sides was in flames, as armies burned enemy settlements and captured forts only to abandon them soon after. The War of 1812 was at an impasse by 1814. No land campaign could succeed there, Wellington mused, unless dominance of the lakes and rivers could be assured.
Britain focused its punitive campaign on the Atlantic seaboard, where its naval superiority could not be challenged. Britain's goal was not conquest, but to punish America for its opportunistic war and to force the U.S. to the negotiating table. Its finances in ruins, Britain also wanted the restoration of trans-Atlantic trade. Ending the war quickly was its primary objective, with the recent shadow of the Napoleonic Wars looming nearby.