NY State Kdg. teacher here. I won't tell you what to do b/c I don't know your child. Whether to hold back or not is a huge issue lately on the DIS boards! It's a tough call but you need to trust your "gut". Kdg. readiness depends on each child.
This year, I had some students who attended pre-K and some who didn't. I'm happy to say that they are all doing well and are on-target for first grade. They're a bit behind in the reading ... but that has to do w/them having a difficult time sounding out and blending words. Many parents do not help w/studying sight words or practicing reading/sounding out. I can only do so much in the time that I have! Anyway, those who didn't attend pre-K were a bit behind the others but they all managed to catch up.
I can tell you that the pre-K kids knew their alphabet letters out of sequence, could recognize numbers through 10 out of sequence, knew shapes, colors, could count to 50+, knew their name, address, phone number, parents names and birthday (month/day), could print their first name and knew how to behave in the classroom. Those who weren't in pre-K learned those things along the way and had a bit more "catching up" to do ... but managed to do it.
Personally, I feel that pre-k is very important. It's a step I wouln't miss out on. But, that's just me. Kindergarten has a curriculum that must be fulfilled and it's less socialization and more education. Having Pre-K skills helps w/Kdg. readiness ... socialization, classroom behavior, etc.
You know your child. I held my child back even though he made cut-off by a month. I just knew that, at age almost 5, he wasn't ready for Kdg. It was the best decision that I made. The extra year of pre-K really put him ahead of the ball-game. He had the self-confidence and skills that he needed. At age almost 5, he would've done "ok" in school and it might've been a struggle. I wanted more than "ok" for my child and didn't want him to struggle and then hate school. He's top of his class and he does a great job.
Your DD might be ready ... have her go to the Kdg. testing at your chosen school and ask their opinion! Good luck!