Leave the original ends long enough so you can thread them and stick them down too -- it does use up a lot of thread! I'd recommend something like a ball of fine crochet thread (embroidery floss weight).
Make a wavy motion with your needle inside your batting to basically make a couple of stitches. Depending on how far apart your ties are and whether your thread can be seen or not, you can go a couple of inches, surface/take your stitch, double-knot and then go back in again to the next one/bury threads.
Here's a closeup picture of my "Ugly Tie" quilt, some ten years of hard teen aged use and washing on it after I made it. You can see my big "invisible" stitches (I probably actually used dental floss -- 2nd thought it was leftover heavy duty upholstery thread) in the corners, but not so much the ones in the middle.
Last edited by Iceblossom; 08-08-2020 at 05:54 AM.