Is this problem still as hard as discrete logarithm (modified ElGamal)? I am trying to find a vulnerability or proof for the following problem:

ElGamal part.

Given $g\in\mathbb Z_p$ where $g$ generates $\mathbb Z_p^\star$, select randomly $k\in\mathbb Z_p$ and calculate $h=g^k \mod p$. The public key is $(p, g, h)$ and private key is $k$.

To encrypt message $m\in\mathbb Z_p$, randomly select $r\in\mathbb Z_p$ and publish $(g^r, m\times g^{rk})$.

Additional part

Let $s$ be randomly selected from $\mathbb Z_p$. Publish $k+s$ and $g^{rs}$.

Problem

If we know $k+s$, $g^{rs}$ and public key $(g,g^k, p)$ is it possible to get $k$, $s$ or $g^{kr}$?

I have found this article (Is this problem same as discrete logarithm?) which is similar, but I cannot find a way it could help me with my problem. 

Is this construction easily breakable? Can it be proved by transformation to discrete logarithm problem or other crypto problem?
Yes, the system is easily breakable.  We have:

$$(g^r)^{k+s} / g^{rs} = g^{rk}$$

You don't list $g^r$ in your problem statement, however it is in the ciphertext, and so we can assume the attacker knows it.