That little pup doesn't look like it was ready to be separated yet, it doesn't have any roots.

How long have you had this plant? Would you be able to add a pic of the whole plant?

It was part of this plant. Bought it about a year ago. When I repotted, I took one of the newer leaves on the left side and placed in its own pot.
After planting that one, the base of the main leaf rotted and pulled off.

Is the pot painted? Plain terracota are best as they dry
out much better than plasic or glazed clay pots.
The splits look like shipping damage and should not be a problem but they will not heal over. The pup looks too far gone, but it doesn't hurt to try. Use rooting hormone on it
give it bright (without direct sun) light and very very careful with water. Good luck

It was painted, but today I repotted (again) in a plain terracotta with extra perlite to help against root rot. Thanks for the help everyone!

Here's a pic in the new pot.

As to the little pup, if you squeeze gently at its base, is it mushy? if so, that's rotting & forget it, if it's firm & smells OK (not like rotting), cut off that bit of discolored, dark stuff, sprinkle cinnamon on it (at that end), shake off the excess & pot it in the tiniest clay pot you have & DO NOT WATER.

If you have no small pots, you could stand the pup at the edge of one of these growing pots, & let it lean on edge of pot 'til it roots but don't water near it or that'll rot it for sure.