Do you want to save the vine?

If not cut the stems that go into the ground.

If it is large you may have to do it with a bow saw.

Then you can let some suckers grow, or just keep cutting them off

In my experience a hedge trimmer gets the small stuff, then stalls and you have to get the big lopping shears, then back to the hedge trimmer, etc. No easy way. "cut the stems that go into the ground..." It'll be back next year with a vengance if it's the Japanese variety (most honeysuckle in the US now is I read?) that we have here.

Good luck,

I have not actually tried this myself, I just use loppers to get the stuff at ground level as suggested above. However, I have heard that you can dip those loppers in concentrated roundup before you cut to slow regrowth. I certainly wouldn't suggest hedge trimmers.