I had a similar experience in the spring. I had to cut down one of my beloved plumerias to just a few short tips, one of them was less than 4 inches. I sprayed them with hydrogen peroxide hoping it would help stop the rotting. I potted them in perlite and pebbles. (I always root my cuttings in perlite and pebbles with great success.) Surprisingly, even the tiny cutting has grown leaves. I would occasionally water with 1/2 water and 1/2 hydrogen peroxide. I will take photos of my cuttings tomorrow. You are in El Monte, so you still have the advantage of heat to help root. Best of luck!
The four small cuttings in the front are from my Divine plumeria that rotted similar to yours. The tiny one is only a couple of inches and still hasn't rooted but does have some new growth. I don't know if it will make it but at least I have three cuttings that look decent. The back three are healthy cuttings I purchased and rooted this summer.
