Well, it turns out that the B-side of Kirika's character song album for Symphogear G is a song called 手紙 "Letter". It wasn't used in-show, at least not during G - it doesn't make much sense as a battle song, and Kirika didn't do any singing outside battle (ORBITAL BEAT cover in episode 5 notwithstanding).
The lyrics for the song include three segments that are enquoted (in 「」), and have part of the structure of a formal Japanese letter, beginning with the salutation haikei. I think that it is reasonable to construe these enquoted segments as being the contents of the letter that appeared in the special episode (zoid9000, Symphogear fan extraordinaire, agrees).
Alas, it is nightmarishly difficult to render those segments into English, for the reasons zoid9000 points out in his translation of the song (tl;dr it's really Kirika-esque). His translation is a bit stolid for a song sung in Kirika's voice, but I doubt that I could really do any better, so I'm just going to mirror the enquoted segments over here, which should give you a good feel for how it plays out:
「dear everyone, um, what should I say?
Thanks for giving me a place to hold on to with these hands that’ve gotten chilly. ☆^(o≧∀≦)o
I pray that one day this whole world will be happy, then let’s laugh! 」
「Daer god, um, hello it’s nice to meet you.
If I am to disappear, I leave everyone in your hands.ヽ(>□<)ノ
Please plant a big garden with flowurs that spell out the words “thank you”….」
「Dear Everyone, something like this is…well,
This is something I’m no good at, but if I can just pass it on
Rainbows, the wind, time, the sun, all that encompassing warmth
I really love it…I truly loved it very much (p〃Д〃q)
I pray that one day this whole world will be happy, then let’s laugh! 」
zoid9000 explains his translation choices in another post, though I think that it's still going to be difficult for the non-Japanese-reader to fully understand everything that's going on here.
Anyway, it is not difficult to see why Kirika might have been embarrassed if Shirabe had read this.