Hi,
You sure it is not a simple "killed by the frost" case rather than infection?
Because killed by frost is what you seem to describe???
Its probably a case of digging it out and getting rid that way.

That sticky brown or black ooze is the results of a bacterial infection, but the bacteria only grow on dead tissue, usually frost killed tissue. Every time we have a severe frost here in the desert, the saguaros develop oozing black sores, and there is an hysterical drive to run around the desert injecting wild trees with antibiotics, while it has been repeatedly shown that they eventually recover or die regardless of treatment. Give the Cordylines time, and don't overwater them, and any live portions will sprout new shoots, and the rest will dry up into sticks and straw, and can be removed.