You need to choose a language and use it as the basis to define your Kolmogorov complexity for all strings. Otherwise, as you said, the definition of Kolmogoro

You need to choose a language and use it as the basis to define your Kolmogorov complexity for all strings. Otherwise, as you said, the definition of Kolmogorov complexity becomes pointless and vacuous - every string has "Kolmogorov complexity" equal to 1! Once you have done this and your language is Turing complete, your language's program sizes can be transformed to other languages' and vice-versa by simply emulating languages in each other (which does not necessarily give you the shortest possible program, but just a bound for its size).