nasty sticky, gummy substance that oozes from vine
What is that nasty sticky, gummy blood substance that oozes so freely from the vine. And what is best used to wash it off my person? - EWW! As much as I have enjoyed the fragrance of jasmine blossoms, the shared front hedge and two bushes on my mom's property have had too many years of lack of attention. It seems almost as if they were to be let alone they would have this whole neighborhood "wrapped" up tight. One day I might find a person that walks on the sidewalk all wound up like a treat in a spider's web. What can be done to get them back in order w/o completely removing and starting over? Again - EWW - with the sap!
Your Jasmine likely has a Scale infestation.
Honeydew is formed by the insects feeding.
You can treat with Neem Oil. You will likely need to treat weekly until the infestation is controlled.
Here are some links for you.