How thick is the board material, and what material is it? The
propagation velocity depends on the permittivity (dielectric constant)
of the material. The impedance depends on the material and the
spacings and trace widths. Presumably there's a ground plane behind
the microstrip lines (else they aren't microstrip). There are many web
sites that will let you play with microstrip designs, and some that
will give you the response of a coupler like you're describing. Do a
web search for things like "directional coupler" and "90 degree
[microstrip] hybrid". But if you plot the coupler's coupling versus
frequency, you'll find it's zero at DC, increasing to a fairly broad
maximum when the freq makes it 1/4 wave long (accounting for the
velocity factor), and falling again to zero at twice that frequency
where the line is 1/2 wave long. That pattern repeats. If you account
for the response, the coupler is useful over a broad range of
frequencies, as the directivity stays good even as the coupling
decreases (if it's accurately made). You can extend the frequency
range (make the peak even broader) by "tapering" the coupling. (Easier
to see in a picture than trying to explain in words...basically a
cascade of sections, with the center one coupled most closely.)

A point to note: if you make the coupled line say 5/4 wave long at
10GHz, it will couple nicely at 10GHz, but you only have to move by
2GHz in either direction to hit a null at 4/4 and 6/4 wave long for the
same physical line length. But if you make the coupled line 1/4 wave
long, then you don't see a null till 20GHz, and the coupler should be
quite useable between 8 and 12GHz. You can make the coupled section
short by leading the ends to the 50 ohm load and the diode detector
away from the coupled section, at right angles to it, so you don't have
to worry about the length of the resistor and the diode adding in some
difficult-to-calculate way to the overall length. Perhaps they already
are done that way, but from your description that's not clear to me.

Cheers,
Tom