Britain was seeking peace since the outset of the war. The war wasn't Britain's idea and it wasn't something they wanted to pursue; the war was declared on Great Britain by the United States. I've gone into detail about the causes from the US point of view here, and about some other issues in this thread.

Of course, while Great Britain desired an end to the war, they wanted to end the war in a way that left them better off for it, if possible. By 1814, both sides were seeking, essentially, for a victory in battle of such proportions that they could flex a little, and prove to the other side that, if fighting continued, they would lose.

Both sides came pretty close, but the US hung in there doggedly. After the defeat of Napoleon, the possibility of bringing troops over from the continent to North America was something they hoped to use for leverage. But after the British defeat at Plattsburgh and Baltimore, Britain sought to end the war by restoring the status quo ante bellum. The peace treaty was signed on Christmas Eve, 1814.

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Don Hickey is probably the best source for the politics - at least on the American side - of the war, and his book 1812: A Forgotten Conflict is a good general starting point for the war as a whole.