Translation:Today is January first.
I wouldn't call the 10th "special." The counters for days are, for the most part, the general couners (一つ、二つ、etc.) but with "ka" instead of "tsu" (ここのか instead of ここのつ、 いつか instead of いつつ、etc.). The general counter is とお (no "tsu"), so you just add a "ka" to the end of it.
You're not, I don't see how that's learning English. This is American English, and I live in Canada and that's exactly what I say. There can only be one default answer, so making it "today is January first" is the correct translation, but there are still other accepted answers. You can still write it your way of saying it, and it will probably be accepted. If it's not, then just report it as an acceptable answer.