Are all vertices within a strongly connected component with 2 or more vertices part of a cycle?
Suppose it's not the case, it means that there is a node $v$ that is either isolated (contradicts $n \geq 2$) or has an arrow that is incident to it. Let this arrow be from $v$ to $v'$, with no loss of generality. Since we have supposed that $v$ is not part of a cycle, it's impossible to go from $v'$ to $v$, which contradicts strong connectivity.