Kolmogorov complexity vs purposefully inefficient Turing machines
You are stating the theorem incorrectly. The theorem states that the Kolmogorov complexity is the same up to a constant for essentially optimal description languages, which are those in which you can describe running a Turing machine on an input $x$ using $|x| + O(1)$ bits. Given this definition, it is a simple matter of playing with definitions to see that all such description languages result in the same notion of Kolmogorov complexity, up to an additive constant.