I'll be shot down again I think, but what I did to begin weaning the DTs was to go away for the weekend. They were not left without as they had daddy, milk in the fridge and they all had a great time. DH said that the first night was hard, but the babies actually took to bottles and cups because there was no alternative. I don't know if it would work for you. The thing is sometimes we allow our hearts to break for our little darlings who are simply getting what they always want, their own way.
If she'll only have from an open cup, then maybe that's the way to go (though I can imagine it's really frustrating to have to keep guard over the drink). Have you tried different cups like the one that looks like a water bottle, different teats etc.? I'm sure that you have and don't really know what else to suggest. I'm going away for a weekend at the end of June and that will be the last time I breastfeed. A little sad and strangely liberating at the same time.
I wish you luck and hope that you get advice that works, be it from me or someone else.