I was pretty well in the dark during my initial watch of this anime. One thing I have learned through researching this anime is that it requires a bit of outside knowledge in order to grasp the nature of the plot. I liked watching this quite a bit and I just didn't want to be left without a clue while enjoying it.
After I finished watching the 2nd OVA that started airing, I've been doing a bit of digging around and found some very valuable information regarding this series. I had a feeling there was something familiar about this anime on my first run through, but it was in the back of my head and it just didn't seem to come to mind.
The plot is very similar to something from Alice in Wonderland and Through the Looking Glass. I have not read either of these, but I have watched the movie that Disney produced as a kid so some of these concepts did make sense to me once I knew where it was from, but I was still unfamiliar with a few details.
If you want a more detailed explanation, I have found a VERY useful comment while searching around a MAL and icantfeelmyarms pretty much nails it when it comes to describing this anime. This is a direct copy/paste so read it as if he was responding to someone. This is a very large wall of text, but it is definitely worth reading if you are unfamiliar with the story.
Well, "Through the Looking-Glass, and What Alice Found There" is written by Lewis Carroll, who is none other than the man who wrote "Alice in Wonderland," and "Through the Looking-Glass" is the sequel to the story...
It shares a lot of similarities with the "Alice" stories, because the whole theme of both Kyosogiga and "Through the Looking-Glass" is about a young girl who finds herself in a mirrored world. In Lewis Carroll's book, though, Alice is in a world that's a completely mirrored world from Wonderland, while Koto's world is, like you said, a mirror of her home of Kyoto, Japan ;)
Also, the poem you mentioned is actually a real poem entitled "A boat beneath a sunny sky," which is a poem read at the end of "Through the Looking-Glass" that is an acrostic (type of poem, in case you don't know) that spells out Alice Pleasance Liddell, who is the girl Lewis Carroll based his "Alice" books on. This is basically giving us the hint that Kyosogiga was inspired by Lewis Carroll ;P Now, the two stories, Kyosogiga and "Through the Looking-Glass," aren't exactly the same. It's not like the creator of Kyosogiga meant to make Lewis Carroll's story into an anime. However, it is highly inspired by the book.
And, now I'll cut to the chase xDD I don't know if you know the story of "Alice," but it's about how Alice follows a rabbit down a hole, and in order to return home she must follow this rabbit and "save" Wonderland from the Red Queen. In Kyosogiga, Koto apparently is lost in the place called Mirror Kyoto and remember, her goal is to find the black rabbit and return home. This black rabbit in Kyosogiga, who is that lady at the end (I love when Koto's like, "A lady kissed me!" xDDD I lol'd e-e), can also be seen as the White Queen, who in "Through the Looking-Glass" is the "good queen," and the Red Queen (bad queen from the book) can be seen as Shouko, who's the blue-haired girl that wants to capture Koto and prevent her from returning home. In Kyosogiga, when Koto gets eaten by the robot, you can think of that as the baddy's attempt to prevent her from saving the "White Queen/black rabbit" because if Koto DID go home, it'd mean she would have to save the black rabbit and that would make Shouko not in charge any more...
Also, the twins from Kyosogiga can also been seen as the twins from Alice ;)
So ya, my overall experience with this series has turned out a lot better once I got a grasp on what was going on and watching this anime again with this knowledge made the anime a lot more enjoyable for me. It bumped my score of the first ONA up to a 9.