Let $\Omega$ be a bounded, open subset of $\mathbb{R}^n$ and suppose $\partial \Omega$ is $C^1$. Assume $kp=n$, $1\le p < \infty$ and $u \in W^{k,p}(\Omega)$. Then $$\lVert u \rVert_{L^q(\Omega)} \le C \lVert u\rVert_{W^{k,p}(\Omega)}$$ for all $1\le q < \infty$. The constant $C$ depends only on $n,p,q,\Omega$.
I know the inequality holds for $k=1$. How to extend it to this general case?
For all $q \in [n / (n-k), \infty)$, there is $p' < p$ with $$ \frac 1q = \frac1{p'} - \frac{k}{n}.$$ Now, you can apply the Sobolev embedding theorem to get $$ \|u \|_{L^q(\Omega)} \le C \, \|u\|_{W^{k,p'}(\Omega)}.$$ The embeddings for the Lebesgue spaces directly yield $$ \|u\|_{W^{k,p'}(\Omega)} \le C' \, \|u\|_{W^{k,p}(\Omega)}.$$ Combining these estimates yields your desired inequality.
The case $q < n/(n-k)$ can be obtained by another application of the embeddings of the Lebesgue spaces.