I would clean up the kittens with a warm washcloth and then make a new nesting box for her. Once the kittens are cleaned up, she might change her mind about them. If not, you will need to act quickly so they don't start going downhill. You will need milk replacement and maybe a heating pad.
I would also try to coax her into staying with the babies by continually putting her in the nest with the babies and helping them find a nipple. You may have to spend a few hours with her putting her back in but I think that once they start to feed, she'll get the idea.."oh I'm supposed to do this? yeah this feels right"
Cutting the cords to remove placentas is as easy as using tooth floss tied up about 1 1/2 inches away from the belly button. Do this as gently as possible and then with sterilized scissors, cut the cords at least 1/4" away from the floss on the placenta side of the cord.