The Kolakoski sequence (K_n) is perhaps one of the most famous examples of self-describing sequences for which some problems are still open. In particular, one does not know yet whether the density of 1's in this sequence is equal to 1/2. This work, which does not answer this question, provides explicit bounds for the main sequences related to (K_n). The proofs rest on a new identity involving the partial sums of (K_n) and on Dirichlet's pigeonhole principle which allows us to improve notably on the error-term.