Is there a canonical value for the ratio? We could insist all vertices lie in a sphere. This gives the die a round shape and is close to the ratio dice manufacturers use.
Since the vertices are now equidistant to the center we can impose the additional constraint $$n + \frac{h}{2} = \sqrt{\frac{h^2}{4} + m^2}$$ We arbitrarily set m to 1 and solve using Mathematica:
Solve[{k/n == m/(n + h), k/m == Cos[2*Pi/10], w/z == h/(n + h), y/(2*m) == Sin[2*Pi/10], z == Sqrt[m^2 + (n + h)^2], f == Sqrt[(z - w)^2 + (y/2)^2], g == Sqrt[w^2 + (y/2)^2], n + h/2 == Sqrt[h^2/4 + m^2], m == 1}, {k, h, w, z, y, f, g, n, m}] // N
The values are $$k \approx 0.809017 \\ h \approx 0.212332 \\ w \approx 0.285586 \\
z \approx 1.49535 \\ y \approx 1.17557 \\ f \approx 1.345 \\ g \approx
0.653491 \\ n \approx 0.899454 \\ m = 1 \\ \alpha \approx 51.8273^\circ \\ \beta = 90^\circ \\ \gamma \approx 128.173^\circ $$
