They feed preemies with cups--they are bendy, but they are cups. Try a regular cup. Particularly the best, are the Tupperware ones with the flared lip--I have no idea why, but they really are the best at getting little mouths to not spill liquid out the sides.
At about age one is a very good time to give her the opportunity to (quickly) learn, with direct physical guidance from parent(s)!!, how to lift, tip, and swallow from a cup.
You have to be willing to clean up the spills (especially if you forget to watch her and she gets bored=>playtime!), but you'll probably get a lot more liquid into her.
I'm lactose intolerant, so my kids drank juice and water after they weaned. Now I've added the organic lactose free milk, and they like it alright (mostly I just use it for cooking)--but they really like it warmed, with a pinch of vanilla extract. (Those vanilla soy milk people are onto something ;)!)
Another thing to try is soaking her snack-cereal in milk (or juice), not so it's soggy, but just so it's carrying a bunch of liquid with every bite (those chex-type cereals and cheerios-type cereals usually hold up under this treatment). Fruit and some veggies, and all pureed babyfoods, carry lots of water too ...
Good luck :).