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In case of a prime power it is not so hard. Denote R the ring of integers of $Q(\zeta_n)$. First note that $X^n -1$ is separable mod p when p does not divide n. This implies $R [1/p] = Z[\zeta_n, 1/p]$. To check that $R = Z[\zeta_n]$ it then suffices to show that the local rings of $Z[\zeta_n]$ at all primes above p are DVRs. But you have the explicit element $\lambda = 1 - \zeta$ that you defined above; its norm equals to p. This implies that there is only one prime above p and that this ideal is generated by $\lambda$. In particular the local ring at this prime is regular and hence a DVR.