Which should be the angle $\alpha$ for this perfect (triangular) billiard shot?

The key for this problem is to imagine the triangular billiard as having mirror walls that reflect the inner area so as to tessellate the plane. This way, the nine times broken trajectory becomes straight (check it out!):

Now, finding $\alpha$ is a simple trigonometry exercise

$$\alpha=\arctan\left(\dfrac{\sqrt{3}}{4.5}\right)\simeq 21.05º$$