I heard they need 600ml (or the equivalent) of milk each day? Natty drinks 200ml milk in the morning, about 100ml after each of her naps, 1 tub of yoghurt, cream cheese on toast for one meal and bite-sized pieces of cheese with another meal (usually. Sometimes this doesn't happen for one reason or another - sometimes she doesn't want to eat the straight cheese.)
How much is Laila drinking at a time? When she has extra milk, does she eat less of her food?
I would try to introduce the sippy cup like you would when you introduce solids - obviously when she's thirsty she wants a bottle because she knows that works for her, so a sippy cup would just be frustrating, KWIM? Maybe try it with a meal or something? And just keep offering but without pressure. Hope this is somewhat helpful, Linz - of course I'm no expert Hopefully you & Laila will work it all out soon.