How does the cyclic attack on RSA work?
We start with the definition of textbook RSA encryption: $c = m^e \bmod n$. From your first equation

$$c^{e^k} = c \pmod{n},$$ we have that if $c^{e^k} = c \pmod{n}$, then $e^k = 1 \pmod{\phi(n)}$ (Euler's theorem). Dividing both sides by $e$, we get 

$$e^{k-1} = e^{-1} \pmod{\phi(n)}.$$ 

By definition, $d = e^{-1} \pmod{\phi(n)}$. Thus, $$c^{e^{k-1}} = c^d = m \pmod{n}.$$