the St Lawrence River is great to defend Canada, but does not lead into US territory. Lake Champlain is what you need to invade the US, and then advance south along the Hudson (which is what the Battle of Lake Champlain prevented)
The Americans held Lake Erie and a naval arms race was underway (the Americans were even working on a double decker) on Lake Ontario so those are out while Lundys Lane and Niagara in 1814 by that apparently spent American force were a bloody draw as US and British regulars found they were evenly matched (finally after 2 years for the Americans)
The US Army on this front was actually finally a capable and dangerous force, as General Drummond openly admitted.
That massive army has to eat while its in Maryland, and while reasonably prosperous, this region of Maryland is not the breadbasket found further north. It was mainly tobacco plantations and supported a decent but not overly large population. As to state governors, Cochrane had been raiding Maryland shore villages and farms for a year already, so the locals are getting pretty angry about it already.. but angrier at the British.
So once the British have Baltimore and Washington, then what? Neither is the most important region of the US in terms of taxes, food production, productivity or wealth. Those would be either further north or further south. Philadelphia and the Delaware River valley, or New York and its environs, or Boston for what served as industry at this date as well as wheat and corn, while further south for the wealthier cotton and tobacco plantations (and wealth).
So once this army eats everything up, while under pretty steady harassment (which had already become a problem for British naval parties)
As to the capital, Congress and the President escaped and would easily make for either Richmond or Philadelphia. Most of the records made it out too. The British burned some buildings and a lot of furniture plus a nice naval yard when DC fell. But that was it. Ross complained about lacking cavalry and artillery. While this is a region known for just about every gentleman being an excellent rider, and Virginia is right next door. Not up to charging a British square but ideal for harassing foraging parties and wagon trains.
So this army isn't going anywhere.
It is also notable that the Hartford Convention and New England calls for secession ended after the burning of Washington, because they felt it was their capital too.
And then a few months latter, should this army stay through the winter, Napoleon escapes from Elba and the British have traded Maryland for a renewed threat from the "Monster"
And still haven't conquered the US or likely brought it to terms.
Meanwhile Brown and Scott have recovered, still have a decent army in New York, while there is no threat to New Orleans (a far more important place) and what have the British really gained? Other than occupying the north woods of Maine and part of Maryland, burned down Baltimore (Cochrane really wanted that) and Washington, they hold nothing of decisive value. Neither the industrial heartland (such as it is) or the really valuable cotton production territory to the south.
And American privateers are still capturing large numbers of British merchantmen, and the costs of the war continue to go up
A few problems with a heroic continued American resistance, the loss of both Baltimore and Washington is a blow to American pride and paints an (untrue but it is about perception) picture the British could go and seize another city if they so desire. The government has been forced to flee the capital and the British are still tightening the blockade (while privateers are not significantly hampering any British operations). Meanwhile the longer the British stay in Maryland and near Virginia the more the very planter elite are going to want peace since the British would be accepting runaway slaves into their ranks and ruining that plantation system. The American Army meanwhile, despite a bloody stalemate at Lundys Lane cannot launch another invasion of Canada. While there is an arms race on the Great Lakes, the Americans are no closer to winning on Lake Ontario than they were last year.
Meanwhile blockade tightens, there are very few offensive options left to the US and there will be discontent from all sides of the country over the running of the war. What possible reason would Madison have to decide to continue the war?
Also one cannot assume that in this alt 1814 Napoleon would still escape, much less do better than he did historically in the 100 Days. So barring that (which didn't slow the war considerably in North America OTL) it's not correct to assume there will be another distraction in Europe.
Again, this would more than likely force peace, there's far too few reasons to continue the war from the American perspective and the British, who have instead taken an eye to ending the war here, would be seeking probably less harsh terms that at first at Ghent OTL[1], so I again sincerely doubt Madison would decide he must risk a continued war.
[1] Well they might insist on Maine and some of the North West, but the Indian Confederacy idea is still a no go.