I think what can be found in that book is a proof of the coding theorem, which is different from the claim you call the “Levin bound” here, I think. What you call the Levin bound is proved in this paper and has a much simpler proof than the coding theorem.
There is a proof of it in “An Introduction to Kolmogorov Complexity and Its Applications” by Ming Li & Paul Vitanyi.
I think what can be found in that book is a proof of the coding theorem, which is different from the claim you call the “Levin bound” here, I think. What you call the Levin bound is proved in this paper and has a much simpler proof than the coding theorem.