I don’t have kids, so I don’t know how well my advice would be, but I can give this in lew; I can’t blame her. The first one’s made her sick, and the second ones she gagged on. I’d be pretty darn wary of them myself if my first experience was bad and my second experience none the better. It sounds like she’s a little afraid of the effects and maybe of gagging and choking on them.
Like I said, I don’t have kids myself, but I have a lot of experience in being one once, so I like to think I’m able to relate to how they think and feel. Ask yourself if it’s a matter of how you approach the situation; is the situation a matter of how the child is reacting, or is the situation a matter of what the child is reacting to – and which one are you reacting to and how? If the meds made me sick, I wouldn’t want to take them either. And if she’s gagging on them, they may be to big or she may have trouble swallowing them. Has she had trouble in the past with swallowing other pills, such as aspirin? If not, were they smaller than the ones she’s taking now?