We derive new constraints on the mass, rotation, orbit structure and statistical parallax of the Galactic old nuclear star cluster (NSC) and the mass of the supermassive black hole. We combine star counts and kinematic data from Fritz et al (2014), including 2500 line-of-sight velocities and 10000 proper motions. We show that the difference between the proper motion dispersions sigma_l and sigma_b cannot be explained by rotation, but is a consequence of the flattening of the NSC. We fit the surface density distribution of stars in the central 1000 by a spheroidal cluster with scale ~100 and a much larger nuclear disk component. We compute the two-integral distribution function f(E,Lz) for this density model, and add rotation self-consistently. We find that: (i) The orbit structure of the f(E,Lz) gives an excellent match to the observed velocity dispersion profiles as well as the proper motion and line-of-sight velocity histograms, including the double-peak in the v_l-histograms. (ii) This requires an axial ratio of q= 0.73+-0.04 for r<70 consistent with our determination from star counts. (iii) The NSC is approximately described by an isotropic rotator model. (iv) Using the corresponding Jeans equations to fit the proper motion and line-of-sight velocity dispersions, we obtain best estimates for the NSC mass, black hole mass, and distance M*(r<100)=(8.94+-0.31|stat+-0.9|syst)x10^6Msun, Mbh=(3.86+-0.14|stat+-0.4|syst)x10^6Msun, and R0=8.27+-0.09|stat+-0.1|syst kpc, where the systematic errors estimate additional uncertainties in the dynamical modeling. (v) The combination of the cluster dynamics with the S-star orbits around Sgr A* strongly reduces the degeneracy between black hole mass and Galactic centre distance present in previous S-star studies. A joint statistical analysis with the results of Gillessen et al (2009) gives Mbh=(4.23+-0.14)x10^6Msun and R0=8.33+-0.11kpc.
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