What to use to kill Asian Jasmine

Trichlopyr is pretty effective, with surfactant added, but you would need the high concentration product, not the 8.8% active ingredient product sold at retail garden centers.

Crossbow (trichlopyr + glyphosphate combo) is reported to be very good. Glyphosate by itself (“Roundup”) is not very effective unless you use multiple high concentration doses (that was my personal experience).

It is a difficult vine to kill.

A sharp shovel

I just removed a bunch. I hit it with weed eater to open up stems then sprayed a vinegar solution to weaken it. From there it's back breaking shovel work. Pry up those roots and pull it up like a carpet! Make sure completely turn the dirt and get it all out so it doesn't try to come back when you lay sod. GL

There aren't any herbicides that are effective on asian jasmine unfortunately. I bought a house that was neglected to the point that it grew up to the second story window and was growing inside the window itself. The only way to remove it is to dig it out. Mine had grown to a foot thick at ground level, so I bought a walk behind string trimmer and used that to cut it down low, and then I tilled and raked and I keep repeating when it grows back. I'm still not completely rid of it, but it is a lot better.