I judged when my son was ready for a cup (without a lid) by watching him every day when I brushed his teeth. He has a tooth mug and gradually I saw he was controlling the cup better and drinking some of the water from time to time instead of just tipping the water down himself! This happened a few months ago and he is currently 27 months old.
With regard to weaning on to a lidded cup, well I know a lot of people wouldn't agree with me, but I thought the anywayup cups were a waste of money because my children never seemed to get used to having to suck to get the fluid out. So I would recommend a lidded cup with an ordinary spout (but with a flip over lid so you can transport full cups when out) as the fluid comes out much more easily.
Just keep trying with it. Your dd will adapt to the cup much less easily if there is always an alternative on offer.