I believe that the twine you tried to dye is really not nylon at all, but instead another synthetic fiber, such as polypropylene or polyester. Can you buy twine from a different source? Sometimes the fiber content of an item is described incorrectly on the label. If you are curious, you might be able to figure out what the fiber is by doing a burn test, but that wouldn't help you with what you need right now, which is turquoise-dyed twine. Using an entirely different twine is probably your best course of action at this point.

If the twine were nylon, it should have picked up at least some of the color. It certainly sounds like you did everything correctly enough that the nylon would have taken some color. Some nylons are treated with finishes that repel dye, but even in most of those cases at least a little dye will be retained, even if the results are not as even as desired.

Polypropylene twine is impossible to dye, as it is completely water resistant, due the the chemical nature of the fiber. Polyester twine can be dyed by extensive boiling with disperse dye, such as "iDye Poly" by Jacquard Products. (Disperse dye can also dye nylon and acrylic to pale to medium shades.) Neither polypropylene nor polyester will take acid dyes at all.

One other possibility is that you might have cotton twine, which cannot be dyed with acid dyes. Cotton twine, if not treated with waxes or other surface finishes, can be very easily dyed with fiber reactive dyes (such as Procion MX dye or Dylon Permanent Dye) in room-temperature water, or even with all-purpose dyes (such as Rit or Tintex) in simmering hot water. The latter will be less resistant to fading due to washing, but this may not be a problem for a one-time use. Just be sure you wash the excess dye out thoroughly enough after dyeing that no dye comes off on other things.

If your twine is not in any way water-resistant, then you can try a thin fabric paint, such as Jacquard Products' Dye-Na-Flow or Dharma Trading Company's Dharma Pigment Dyes. These will work on both natural and synthetic fibers, but only if the fiber has not been treated in any way that provides resistance to water. Fabric paints will work on untreated cotton, nylon, and polyester, but not on polypropylene.

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