When I was looking around trying to find some inspiration to answer your question, I found the following result of Feferman from 1957:
For any set $X$ of natural numbers there is a theory $T(X)$ such that:
The set $X$ and the set of Gödel numbers of consequences of $T(X)$ have the same degree of unsolvability.
If $X$ is r.e. then $T(X)$ is effectively axiomatizable.
Because there are nonzero r.e. Turing degrees strictly weaker than $K$, I think this may answer the question.
The result is in the paper "Degrees of Unsolvability Associated with Classes of Formalized Theories", Solomon Feferman, The Journal of Symbolic Logic, Vol. 22, No. 2 (Jun., 1957), pp. 161-175. http://www.jstor.org/stable/2964178
