So good, I'm going to quote it:
"...From a purely observational standpoint you have gone on ad nauseam about how you have made her smarter, how academically superior she is, nowhere in here have I seen mention of a friend, just one, or playing or anything social." -- Jo W's post, below. Jo has hit the nail on the head here.
You don't mention anywhere in your post the following things:
When you were told she would have to repeat K, did you talk to the teachers about whether it was strictly because of her age or whether it might be because of a need for further socialization and maturity? The teacher "agrees that there would be no benefit to repeating," but is that only "no academic benefit"? Your child would be by far the youngest in a first grade class and the youngest ever after. Smart kids often ARE more mature and she may well be, but that doesnt' mean she's ready for first grade. You need some frank talks with the teachers about whether she might benefit from the social aspect of another year of K and having another year to mature. There can be a huge gap between a five year old and a six year old--even if the five year old is book-smart and has a ton more factual knowledge.
You also don't say that the school ever told you, when you put her into K, that there was a guarantee she would just advance into first after that. Did you ask, when you enrolled her, if she would definitely go into first the next year? If not -- they did not give you any guarantee of that, and it is the school's choice whether to promote her.
Frankly, you set her and yourself up for this disappointment by entering her in K so early and making assumptions about what would follow. I don't say that to be mean, I say it in hopes that you can be objective enough about what you've done to see that you didn't think through what would happen next.
Can you be objective here? Can you step back from your very, very, VERY high expectations for this little girl and see that she needs to be a little girl who is well-rounded and not just on a fast track academically? That school in the early years is as much about maturity as about academics? And that she may feel very pressured by your expectations? Even if you do not ever tell her she's brilliant, or point out to her that "You went to K early!" etc. she is probably still totally aware that you expect her to be perfect academically. It's a big weight for a young child, even one who just loves to learn. You surely will say, "No, I don't expect perfection!" but that is how a child her age will see it. If she doesn't go into first next year, she has let you down. She has done wrong, done something bad, hurt mommy. That is how she will see it. Because she is five. Can you think at her age level and see how her emotions may work, not just how her brain works or how she talks more maturely?
If you think she'd be bored, pull her out of this school for a year. Travel with her, take her to museums, but most of all get her into playgroups, have play dates, join kids' classes that are not academic (tumbling, dance, sports, art) and get her a LOT of exposure to other kids of all types, not just super-academic ones. Then start her in first on time.
I predict that when she does hit first grade you will probably find other problems with her schooling -- you'll feel that it's not advanced enough, the other kids aren't at her level so she's not with a peer group that pushes her, etc. You want the best for her but are you setting her up for always feeling like she's the odd one out because her parents feel whatever setting she's in isn't good enough?
Yes, as someone noted, you could homeschool her if you prefer, but please get into some good homeschooler groups where she can socialize a lot. Meanwhile, have frank talks with the current K teachers. And talk with the first grade teachers, who may have some real reasons they don't want her in first grade yet. You have to ask them even if you don't like the answers you may get. If you just dump the school and never ask, you may miss out on some good advice they could give you.