Gorgeous string quilt! Your use of value makes those blocks look like gemstones!
I really like the look of that rug as a border too! If you could find a dark gray fabric like that it would look great. If it were mine I would be tempted to put a dark gray or black border of about 2" finished size then a very, very narrow border of your pieced rainbow colors blending from one to the other. Yellow shading to orange, shading to red, shading to purple shading to blue shading to green which can shade back into the yellow. I would piece the strings then cut them so they finish no wider than 1/2" then repeat the dark gray black border at around 4" wide.