Try bleach. Get a sponge wet with water, add some bleach (about 1tsp to start, more if it isn’t strong enough) to it and rub that on your nails. Once you are finished, rince your hands with vinegar so they don’t get a bleach burn, then use regular soap and water to get the vinegar smell off.

Usually there is more dye on the cuticle area as well. You will need to use a nail brush and possibly cuticle remover to clean up that area. Bleach won’t usually remove the dye from the dead skin. If it is superficial you could try buffing it off of your nails. And finally, if all else fails, cut them shorter and keep them polished until they grow out. Good luck.