Kansas probably doesn't change much: same borders, but date of admission a year or two later. The statehood petition was already well underway when South Carolina seceded (submitted in 1859, approved by the House in 1860), and the supporters of the anti-slavery Wyandotte Constitution were firmly in control of the territory. Without secession (and the withdrawal of most of the Senators from the seceded states), there wouldn't be a majority in the Senate to approve Kansas statehood in January 1861 like IOTL, but the margins were pretty slim, especially after the new Congress was seated in March.
Assuming OTL election results, Republicans would be two seats short of an evenly divided Senate (with Hamlin as tiebreaker) when the 37th Congress convened. And even assuming no Democrats would vote for Kansas statehood (not sure how accurate this assumption is: not all Democrats were pro-slavery, and even some pro-slavery Democrats might have been inclined to accept the outcome on the ground in Kansas), IOTL two northern Democrats (Douglas (IL) and Thomson (NJ)) died during the session and were replaced with Republicans.