I've been thru your pain but with a different vine. Don't think the battle is over yet. I'm still in a wait and see mode.
Do NOT use landscape fabric. Won't help. The mulch will be protecting the soil during this painful process. Mother Nature does not like to be naked. That's where the mulch comes in. And good mulch (shredded native hardwood bark) will eventually provide nutrients to the soil.
I personally would not use Round-Up. You don't say what Zone you're in or where you're located. If you are still getting hot weather you can try using 20% vinegar (some say 10%.) Not something you would be able to find at the local grocery. And something to be very cautious using - wear protective clothing! Anyway, applying vinegar on the leaves, and maybe on the roots if you can see them, during a sunny day will kill - but I think it needs to be multiple days. Use a spray bottle. Some people have used a little paintbrush to apply the vinegar. You don't want to harm your soil any more than necessary. (You can google vinegar as a weed killer and see if there's any mention of ivy or vines.)
How about a picture or 2 so we can see your progress?
Here's a picture of the 'remains of the day' that I've almost eliminated. Haven't had enough direct sun to try the vinegar on any remaining roots. I've been using little yellow flags to mark where I've pulled up the roots that you'll see in the picture. I want to be sure that none of them are going to live and I want to know where they started, if this makes any sense. Sorry. I'm getting worked up. It's war! But it is time that I get some good mulch down or I will be growing mud! Not going to plant anything else for awhile but that has more to do with location, etc.
