As has already been mentioned, the Electoral College has already convened and cast their votes for Pierce. What has not been mentioned is that on the same day of this accident, Congress will be convening in a joint session to count and verify the results. If the President-elect's death spreads to Washington before or during the count, under 3 U.S.C. § 15 any sitting member of Congress may issue an objection to a state's vote because the "results [have not been] regularly given." In this situation, one could argue that legally any votes received for Pierce are in violation of law, because you cannot, after all, elect a dead man.
While it would require exceptional knowledge of procedure to be acted upon, the opposition have an opportunity to sow chaos within the outgoing Congress. So long as the motion to object is seconded by a member of the opposite house, the joint session is suspended and both houses are forced into a special session to debate and vote upon the matter. With representation in both houses and Charles Sumner among their rank, the Free Soil Party is just the group to exploit parliamentary procedure in such a way. Doing so forces moderates in both parties to make the explicit decision to support the radically pro-slavery Atchison as President-Elect, which many will find anathema. The Free Soilers can also threaten to "run out the clock" in which they filibuster until the new Congress is supposed to be sworn in and trigger a Constitutional Crisis.
Everyone in Washington knows that King is not long for this world and so a vote to verify the results becomes a vote for Atchison. In an absolute worst case scenario, where the Free Soilers manage to lead an anti-Atchison coalition to invalidate election results, you could very well trigger the Civil War seven years early. Unfortunately, the Whig Party is on the verge of collapse, there is no strong leadership anywhere in the country, the fight is no longer explicitly about slavery, and neither side is really prepared.