The characterization given by Noam Elkies is correct. Indeed, in this paper Bryant, Horsley and Pettersson prove the stronger result that if $n$ is odd, and $m_1, \dots, m_t$ are such that $m_i \geq 3$ for each $i$ and $\sum_{i=1}^tm_i=\binom{n}{2}$, then the edges of $K_n$ can be decomposed into $t$ cycles with specified lengths $m_1, \dots, m_t$. If $n$ is even, and $m_1, \dots, m_t$ are such that $m_i \geq 3$ for each $i$ and $\sum_{i=1}^tm_i=\binom{n}{2}-\frac{n}{2}$, then $E(K_n)$ can be decomposed into a perfect matching and $t$ cycles with specified lengths $m_1, \dots, m_t$.

The characterization given by Noam Elkies is correct. Indeed, in this paper Bryant, Horsley and Pettersson prove the stronger result that if $n$ is odd, and $m_1, \dots, m_t$ are such that $3 \leq m_i \leq n$ for each $i$ and $\sum_{i=1}^tm_i=\binom{n}{2}$, then the edges of $K_n$ can be decomposed into $t$ cycles with specified lengths $m_1, \dots, m_t$. If $n$ is even, and $m_1, \dots, m_t$ are such that $3 \leq m_i \leq n$ for each $i$ and $\sum_{i=1}^tm_i=\binom{n}{2}-\frac{n}{2}$, then $E(K_n)$ can be decomposed into a perfect matching and $t$ cycles with specified lengths $m_1, \dots, m_t$.