
Let's see. That's a good question. Probably the size/gauge of your jump rings will be determined by the size of those double chains you're using and by the size and weight of your clasp, especially if that's where you're putting your jump rings (i.e., right there connecting the chain or chains to the clasp). You're going to want your jump rings to blend in and look close in size to your clasp and chains.
Probably you'd want to get a few bags of jump rings in different sizes and just see what matches up best to your chain and clasp. In the photo to your right, I'm showing you some sizes (from left to right): 12mm, 9mm, and 6mm. The split rings are on top and the jump rings are on the bottom. The jump rings have a more delicate look, but the split rings offer a little more security. You could also consider soldering the jump rings that you use. That's probably what I would do.

Then there is always the possibility (and my apologies in advance if you already know all about this) that the wrong technique in opening the jump rings is damaging their structural integrity. Opening a jump ring by pulling the ends straight back and keeping them in the same plane will damage it. Opening it as in the photo (twisting the ends in two different directions and moving them into different planes) will enable them to be twisted closed again with no problems.
Hope this helps (also with you, too, Angie)! Feel free to post a photo of those chains/clasp you're working on and we'll all try to advise you from there. Good luck!