I would also have put it in the pile.
Probably is taking a while to break down because the vine is so sturdy. When we put small cherry laurel branches in our pile, we see them for a while, too.
Just keep turning the jasmine back into the hottest part of your pile and they will break down.
One other idea, we have an electric hedge trimmer that I bought before I realized I had no intention of leaving our bushes as big as they were.
So, when we pulled up our pole beans and our okra, I put it all in a big pile on the ground and used the hedge trimmer (kind of like a big electric knife) and just sliced down through the pile to the ground. After a dozen or so cuts, everything was in smaller chunks, was easier to put in the pile and -- I think -- was easier for the pile to break down.
Happy gardening!