I am simply concerned with the final word, which is "Republic", where the Law rules the People.
Adding "democratic" as a qualifier to "Republic" seems to confuse many more than it informs, in my experience. As it stands, the majority of people I encounter thus seem to falsely believe this is a Democracy, and from the way they describe it to me, they also equate that word with "Freedom".
In any case, it's rarely democratic at all. A great portion of the regulations governing citizens actually come from unelected administrators in various executive-branch agencies who enact policies to rule over their subjects among the territories of the United States. In this sense, I would argue that if you do feel pressed to use a qualifier, the term "oligarchial Republic" might be far more accurate instead.