  We consider the one-parametric family of self-adjoint realizations of the
two-dimensional massive Dirac operator with a Lorentz scalar $\delta$-shell
interaction of strength $\tau\in\mathbb{R}\setminus\{-2,0,2\}$ supported on a
broken line of opening angle $2\omega$ with $\omega\in(0,\frac{\pi}{2})$. The
essential spectrum of any such self-adjoint realization is symmetric with
respect to the origin with a gap around zero whose size depends on the mass
and, for $\tau < 0$, also on the strength of the interaction, but does not
depend on $\omega$. As the main result, we prove that for any $N\in\mathbb{N}$
and strength $\tau\in(-\infty,0)\setminus\{-2\}$ the discrete spectrum of any
such self-adjoint realization has at least $N$ discrete eigenvalues, with
multiplicities taken into account, in the gap of the essential spectrum
provided that $\omega$ is sufficiently small. Moreover, we obtain an explicit
estimate on $\omega$ sufficient for this property to hold. For
$\tau\in(0,\infty)\setminus\{2\}$, the discrete spectrum consists of at most
one simple eigenvalue.
