Finite length for unramified ${\mathrm{G}\mathrm{L}}_2$

Let $p$ be a prime number and $K$ a finite unramified extension of ${\mathbb{Q}}_p$. If $p$ is large enough with respect to $[K:{\mathbb{Q}}_p]$ and under mild genericity assumptions, we prove that the admissible smooth representations of ${\mathrm{G}\mathrm{L}}_2(K)$ that occur in Hecke eigenspaces of the mod $p$ cohomology are of finite length. We also prove many new structural results about these representations of ${\mathrm{G}\mathrm{L}}_2(K)$ and their subquotients.