Edward
Kilim rug fringes are usually cotton and although some dont like this process as it reduces the fringe life, I use it quite a lot as cosmetically it makes the fringe look white again. Fringes usually last over 10 years so an old expensive rug is worth refringing.
I would use the Prochem 3 chemical mix, Fibre shampoo, oxibrite and fibre buff. Take care not to get any of the mix onto the rug its self though as it will strip dye colour from it.