Seeing as she startled a gazelle herd when she tripped over a rock she tried to avoid and couldn't keep up with the running herd, she would have struggled to catch her prey if the fire outbreak didn't occur. Plus, when Kovu was teaching her to hunt, he told her that she was breathing too hard (and was making noises), so the herd would have heard this, and ran. She probably would catch her prey, but it would have took longer than Nala or the matured lionesses would take.
It's obvious that Simba's over-protection played a role in Kiara's hunting skills. It's otherwise confirmed that Simba makes judgement on it when he yells, "No more hunts for you! Not ever!" As hinted in the first movie when Scar summons Sarabi, the Queen of the Pride Lands is the leader of the hunting party. So yes, Kiara's hunting lessons fell on Nala, but Simba's over-protectiveness hindered her from being an accomplished huntress. I'm sure Nala taught her well, but Simba stood as the barrier from her hunting skills to improve. And like DGFone stated, this opened the door for Kiara and Kovu to develop their relationship at the expense of Nala's character.
One theory I have to add to the table as to why her hunting skills were below par is that her build (and her body length) wasn't fully developed yet. In this
screenshot, she only comes up to Nala's shoulder, and her body length isn't as long as her mother's. Director Darrell Rooney did mentioned that Kovu was a "teenaged killer" in the
Proud of Simba's Pride documentary, and Kiara being around that same age-span makes sense as
Romeo & Juliet serves as the inspiration for the storyline in SP...and the title characters are, you guess it, teenagers. Meanwhile, I can assume adult Nala in the first movie was much older than "adult" Kiara so her body was capable of the challenges of hunting.
So, in conclusion, I believe Kiara can be a much better hunter now that Simba doesn't have to be over-protective of her now and that her body needs to mature more.