He wins--the Whigs are just too split on the slavery issue to prevail in 1852, and the prosperity following the discovery of gold in California has rendered their economic program pretty much irrelevant.
As president, Cass goes along with the Kansas-Nebraska bill. The pressure from powerful Democrats in Congress (Douglas, Atchison, Mason, etc.) is probably too much for any Democratic president elected in 1852 to resist--except Sam Houston, and his chances of getting the nomination that year are dim. Cass was, after all, the father of "popular sovereignty" (in the 1848 campaign) and would find it hard to argue that the principle he had advocated for the Mexican Cession should not also be applied to Nebraska.
From there, I'm afraid things go much as in OTL. Anti-Nebraska forces make big gains in the 1854 election. In 1856, the Democrats dump Cass in favor of a "safer" candidate, like Buchanan...