I apologize for this delayed reply to your gardening question and hope it is still helpful to you.
Your options are to either dig it out or possibly to mow it to the ground and then cover it with a landscape weed block fabric which would then be covered by mulch. You can plant through the fabric later on if you like and it should do a pretty good job of shading out an killing the jasmine as it is trying to regrow. I am not aware of any natural product that would move into the plant and kill the entire plant, roots and all.
Thanks for the question. Best wishes for a wonderful gardening season. Please stop in again soon!
Q&A Library Searching Tips