Running time analysis for algorithm that merges $k$ sorted arrays to one sorted array
You are almost there. In the first merge pass we do $\frac{k}{2}$ merges each taking time proportional to $2n$. In the second merge pass $\frac{k}{4}$ merges each taking $4n$ (note how nicely the factors 2 and 4 cancel and each merge pass runs in $\Theta(nk)$ time). Finally, since we need only $p = \log_2k$ merge passes, we obtain the running time of $\Theta(nk \log k)$.