What if there had been a 2000 Florida like fiasco in Cook County with Nixon challenging the election results leading to a long and drawn out recount?
There was.
Granted, Nixon bullshited his way through it by saying he was taking a higher road and such, but that was all saving face. Eisenhower didn't give his support to a recount and so Nixon didn't really have a leg to stand on.
And Nixon had his friends and colleges lead calls for a recount and get a nation-wide recount held.
The results: votes for Kennedy were over counted in many districts, but those numbers were negligible and not enough to give Nixon victory. Likewise, in many districts, it was found votes for Nixon had been over counted themselves. All that resulted from the recount was Nixon gaining -3 Electoral Votes; it turned out Hawaii had voted majority for Kennedy.
Granted, you could try to find a way to make it bloodier, but I don't think the support was there. Ike wouldn't back Nixon on it, and the GOP weren't angels themselves; they may have railed against how the Democrats were rigging things, but they were playing the same game with the same dirt. The GOP controlled most of Illinois' counties outside of Cooke, for example, and the Democrats were noting veeery fishy numbers coming out of southern Illinois. You can bet your sweet bippy that if Nixon and the GOP make a huge case out of the recount, the Democrats will pull that card out. They said they would do as much if there was a chance of Kennedy losing.