DS needed one recently (he was 20 mths at the time), the Paed recommended our local public hospital as he knew they saw a lot of young children. I bought some of that emla cream as well jic. I worked myself into quite a state beforehand stressing out over how horrible it was going to be but he was a total champ! He watched The Wiggles on my iPhone while the 2 nurses took the blood, he didn't even flinch and watched the whole thing with great interest. The nurses were so surprised!

My advice would be to call around first and find out which place is best for kids or just take her to the children's hospital. You might have to wait awhile but at least you know they're experienced in taking blood from little veins.