Trying to get rid of well established asian jasmine bed is one tough job and I'm not sure that you'll ever get rid of it completely. I've gone at it with a weed eater, but the stuff I had was a bit much for it. What worked for me was a cheapo electric hedge trimmer to shear it off to the ground - much like Bekah and her husband did.
After that it's back breaking hand and tool work trying to get all of the roots, or as much as you can stand before the urge to call someone else in to do it for you - or any number of worse things.
Depending upon the size of the beds it could be a long weekend's worth of work just to make a dent in it. In my case I'm still not done after perhaps 4 or 5 afternoon's of hard labor.
If no one else has any better suggestions then you might call around to some nurseries to see what techniques they recommend. Good luck!
~Dave