Consider the following construction:
What we do know are the length $L = |AB|$ and the angles $\alpha$ and $\beta$. I want the length $|AD|$ of the diagonal AD.
The only thing I managed to do so far is to get the angle $\hat{AEB} = \pi - \alpha - \beta$ and $\hat{AEC} = \alpha + \beta$. Of course, $|CD| = |AB| = L$ and $|AC| = |DB|$ is still missing.
How to find the length of the diagonal AD?
Firstly, $|AD|=2|AE|$
Using the sine law on triangle $ABE$ ... $$ \frac {|AE|}{\sin(\alpha) }=\frac L{sin(\pi-(\alpha+\beta))}$$ So $$|AD|=\frac{2L\sin(\alpha)}{\sin(\alpha+\beta)}$$