Clarification of Proof: Proving perfect secrecy for One Time Pads
As you have shown, the number of keys such that $E(k,m)=c$ for given $m$ and $c$ is $1$. So the probability

$$Pr[E(k, m) = c] = \frac{1}{number\,of\,keys}$$

with $k$ chosen randomly from $K$ is the same for any plain text and any cipher text, so

$$Pr[E(k, m_0) = c] = Pr[E(k, m_1) = c]$$

for all $c \in C$ and $m_0, m_1 \in M$ with $|m_0| = |m_1|$, and thus by definition the OTP has perfect secrecy.