Try sticking the leaf that fell off into sharp and dry grit in a small pot that will let you rest the leaf against the side. Not too deeply. Simply rest the stem end barely into the grit.

Ignore it for a while. Let it callus over.

It is very likely to produce little offsets, a bit like an African violet. When it does, water once a week until the offsets are big enough to pot up. Unless the other leaves are seriously degraded, you can try the same move with any others that fall off.

(I also threw some fallen leaves into the compost heap and was a little shocked to find the same thing happening a few layers down when I came to turn the pile. Tough as!)

I'd also downsize the pot, for now. The roots are probably mush.

If things get seriously desperate - rescue the top. Snip. Put to callus somewhere in the dry and sun. Wait for the emergence of fine red rootlets. Pot up in a smallish pot with a lean mix - lots of sharp grit. Water sparingly.

I could easily be mistaken: looks like mealie bugs near the top on the right hand side. A wooden tooth pick is often effective for removing these pests.

The leaf that fell off had a discolored portion at the end which was soft. That soft area (seen on the fallen leaf as the dark green section) moved further up the leaf in the last few days until the leaf fell off (I think it happened when I was handling the plant).

Would that leaf still be viable for propagation?

I have cactus mix, perlite and vermiculite. Which combo of those should I use for propagating parts from this plant?

I looked at the plant with an LED flashlight and I did see some white specks on a few leaves, but they seemed to be debris rather than bugs, but, I've no experience with mealy bugs to be sure.

If it does have mealy bugs, the entire thing is likely going into the trash because I'm scared of an infection on my other plants. I've attached another closeup pic that may help someone to confirm mealy bugs or not.

Also, I noticed two little odd "cups" in the soil with little seeds (or maybe fertilizer pellets?) in them.
The cups are attached to a stem which is buried in the soil.
What the heck are those things? (pic attached)

I have a few other succulents, one of which was damaged when it fell. I took all the leaves, left them in a bowl to callus over for a few days, then placed them all ontop of some soil, and so far several weeks later nothing has happened with them, despite them being supposed to root and grow babies... so, clearly I'm not good at propagating plants from leaves that supposedly propagate simply from sitting on soil... so I'm not sure I'd have more success with propagating this plant the same way, though I will try it.

I'm going to take this plant out of the soil today and see what the root system looks like. I'll take some pics and post them here for advice.

At the moment, I just want to confirm mealy bugs or not... the very thought terrifies me.

Here's the pic of the little "cups" and the "seeds".
What are they?

That's bird nest fungus. I would repot immediately. Not that the fungus is harmful, but a sign the soil is probably completely inappropriate (it may not have been originally, but has aged/decomposed to unacceptable conditions for the Kal.)

I've already repotted into a cactus mixture with added perlite.

The soil with this plant was some of the most disgusting I've seen, so I'm not surprised it had that fungus in it.

Glad you did repot! How did the roots look/feel?

This fungus is one of many natural occurrences that can happen in/on decomposing organic matter, not inherently bad or wrong, just a sign organic bits in the soil have decomposed enough to support the fungus. This one is often seen in beds mulched with shredded hardwood.

There wasn't much in the way of a root system.
I gently removed as much of the old soil as possible by rubbing the roots between my fingers and the roots felt very fine.

Here's a picture of the roots.

I just checked the plant and I wiggled it back and forth in the soil and the stem feels firm. The leaves though, are a different matter. One feels very hard, another feels a slight bit more flexible, and one feels soft like it is rotting.
The new small grow on top looks and feels ok.

It was repotted 3 days ago and I gave it a bit of water (not much) and it's sitting about 3 feet from a south facing window.