When we say the sun takes 230M years to orbit the galaxy, what is this in relation to? We measure the earth's rotation relative to the distant stars. What is the reference for our Sun's motion around the galaxy?
The Sun's velocity is known (with uncertainties) with respect to the Galactic centre (GC), as is the Sun's distance from the Galactic centre. These measurements have a long tortuous history, which I am not going to attempt to summarise here. However, the motion of the Sun with respect to the GC is established by measuring the proper motion of the Sgr A* source with respect to the assumed zero proper motions of distant radio quasars (see for example Backer & Sramek 1999; Reid & Brundthaler 2004).
The astrometry frame of reference now is known as the International Celestial Reference Frame and is defined by hundreds of compact, extragalactic radio sources.
If you then assume the Sun's orbit is circular and that Sgr A* is at the GC, then the job is done - the relative velocity divided by the distance to the GC gives the angular velocity $\Omega$, and the orbital period is $2\pi/\Omega$ . If you have a model for the Galactic potential then you can do a bit better.