If $L$ is an intermediate extension of $E$ over $F$, and $\Delta_{N/M}$ denotes the discriminant of $N$ over $M$, then $\Delta_{E/F} = N^{L}_F(\Delta_{E/L})\Delta_{L/F}^{[E:L]},$ where $N^L_F(\cdot)$ is the norm map from $L$ to $F$.
In particular, the discriminant of $L/F$ divides the discriminant of $E/F$.
Let $L = K\cap\mathbb{Q}(\zeta_N)$. Then the discriminant of $L$ over $\mathbb{Q}$ has to divide the discriminant of $K$, and also has to divide the discriminant of $\mathbb{Q}(\zeta_N)$.
The discriminant of $\mathbb{Q}(\zeta_N)$ is $(-1)^{\varphi(N)/2}\left(\frac{N^{\varphi(N)}}{\prod\limits_{p|N}p^{\varphi(N)/(p-1)}}\right).$ In particular, it can only be divisible by primes that divide $N$.