I have seen a proof by contradiction for the uncomputability of Kolmogorov complexity. The idea the basically the same as in the proof for halting problem (i.e., there are cases that lead to Berry paradox example). However, I would argue that since there are infinitely possible ways to interpret a given string, this fact already shows that in the worst case, you can't even iterate through any string, let alone finding the equivalent shortest string for a given string. Is there anything wrong to my reasoning?
Kolmogorov complexity doesn't say anything about "interpret"ing a string or finding the equivalent shortest string for a given string, so your argument is not relevant. I don't know what it means to say that one can or cannot iterate through any string.
Kolmogorov complexity has a precise mathematical definition, as does the notion of computability. Arguments in informal English language can often lead one astray, as they might not correspond to the actual mathematical concepts.
Your comment clarifies that you're concerned that there could be an infinite number of programming languages that could be used. That's not correct. With Kolmogorov complexity, we first fix a programming language -- typically, Turing machines.
I suggest finding a textbook or lecture notes that covers this. Wikipedia is probably not sufficient as a resource, as there are important details that aren't listed there.