Say your bit string is s and the program you find that outputs s is p. Now you know that K(s) ≤ |p|, meaning the length of p. However, we wanted to show that K(s)>L, so the bound is in the wrong direction. We need to show that there's no shorter program that also outputs s, which is impossible by Chaitin's theorem.

The actual proof of Chaitin's theorem is somewhat similar to your argument here. Basically, if we have can prove that any string has a Kolmogorov complexity greater than L, we write a program to search through all proofs until it find a proof that some string s has complexity greater than L and outputs that string. By choosing L sufficiently large, this program itself has complexity less than L, but it outputs s, contradicting K(s)>L.