How to calculate Kolmogorov Complexity if we have access to an Oracle for the HALT Problem
Here is the idea. Given a string $x$ whose Kolmogorov complexity you want to calculate, go over all programs in non-decreasing order of length. For each program $p$, check using the oracle whether it halts, and if so, check whether it outputs $x$. If it does, the Kolmogorov complexity is the length of $p$. Otherwise, go to the next program. Eventually you will find a program computing $x$. The program you will find is a shortest one since we go over all programs in non-decreasing order of length.