How many languages exist over the following alphabets? (a) We have alphabet $\Sigma=\lbrace 1 \rbrace$, $\Sigma=\lbrace a,b \rbrace$ and (b) also an alphabet with size $k$ and words with length $n$.

For the first two alphabets in (a), we know that there are countably infinite words: 
$$|\Sigma| = |\mathbb{N}|$$

For the alphabet in (b) we know we have $k^n$ number of words.

Now I need to find the number of languages that exist for the symbol set $(a)$ and $(b)$.

I think the number of alphabets is also countably infinite, so it has the same size as the power set of $\Sigma$ which has a cardinality $|\Sigma| = |\mathbb{N}|$, therefore there are $2^{|\mathbb{ℕ}|}$ languages.

Now my question is, does this $k^n$ have a role in this $2^{|\mathbb{ℕ}|}$; does it change anything?
A language over an alphabet $\Sigma$ is any subset of the set of all finite strings with elements in $\Sigma$. If $\Sigma$ is finite or countably infinite, and not empty, then the set of all finite strings with elements in $\Sigma$ is countably infinite ($\aleph_0$) and the set of all languages over the alphabet $\Sigma$ has the same cardinality as the real numbers ($\aleph_1$). (Not calling it just "uncountable", because the set of languages is indeed uncountable, but there are uncountable sets with higher cardinality).