$$\delta_{kl}\epsilon_{ijk}\epsilon_{jki} = \delta_{kl} (\delta_{jl}\delta_{ki} - \delta_{ji}\delta_{kl})$$ $$\delta_{kl}\epsilon_{ijk}\epsilon_{jki} = \delta_{kj}\delta_{ki} -\delta_{ii}\delta_{ji} = \delta_{ji} - \delta{ji} = 0$$
What mistake I made here? Should I compute $\delta_{kl}\epsilon_{ijk} = \epsilon_{ijl}$ first because $k$ is not a free dummy index?
Your indexing seems wrong. In the first equation on the left you have three $k$-s, I guess one of them must be $l$.
Next, still first line, assuming the second $k$ is $l$:
$$ \epsilon_{ijk}\epsilon_{jli}=\epsilon_{ijk}\epsilon_{ijl}=\delta_{jj}\delta_{kl}-\delta_{jl}\delta_{kj}=3\delta_{kl}-\delta_{kl}=2\delta_{kl} $$