Answer
$\color{blue}{36.7^\circ}$
Work Step by Step
Using the Sine Law:
$\dfrac{\sin A}{678} = \dfrac{\sin B}{b} \\
\dfrac{\sin 45.6^\circ}{678} = \dfrac{\sin B}{567}\\
\sin B = \dfrac{567 \sin 45.6^\circ}{678} \\
\sin B \approx 0.5975 \\
\color{blue}{B \approx 36.7^\circ}\quad \text{or}\quad \color{red}{B\approx 180^\circ-36.7^\circ \;=\; 143.3^\circ}
$
If $B=36.7^\circ$, then, since $A+B+C=180^\circ$,
$C = 180^\circ-A-B \\
C = 180^\circ-45.6^\circ-36.7^\circ \\
C = 97.7^\circ$.
Thus, such a triangle is possible.
If $B=143.3^\circ$, then, since $A+B+C=180^\circ$,
$C = 180^\circ-A-B \\
C = 180^\circ-45.6^\circ-143.3^\circ \\
C = -8.9^\circ$.
Thus, such a triangle is not possible.
Thus, there is exactly one solution: $\color{blue}{B=36.7^\circ}.$
