  We report a complete set of $^{59}$Co NMR data taken on the $x=2/3$ phase of
sodium cobaltates Na$_{x}$CoO$_{2}$, for which we have formerly established the
in plane Na ordering and its three dimensional stacking from a combination of
symmetry arguments taken from Na and Co NQR/NMR data. Here we resolve all the
parameters of the Zeeman and quadrupolar Hamiltonians for all cobalt sites in
the unit cell and report the temperature dependencies of the NMR shift and spin
lattice relaxation $T_{1}$ data for these sites. We confirm that three
non-magnetic Co$^{3+}$ (Co1) are in axially symmetric positions and that the
doped holes are delocalized on the nine complementary magnetic cobalt sites
(Co2) of the atomic unit cell. The moderately complicated atomic structure
resumes then in a very simple electronic structure in which the electrons
delocalize on the Co2 kagom\'e sublattice of the triangular lattice of Co
sites. The observation of a single temperature dependence of the spin
susceptibilities indicates that a single band picture applies, and that the
magnetic properties are dominated by the static and dynamic electronic
properties at the Co2 sites. We evidence that they display a strong in plane
electronic anisotropy initially unexpected but which accords perfectly with an
orbital ordering along the kagom\'e sublattice organization. These detailed
data should now permit realistic calculations of the electronic properties of
this compound in order to determine the incidence of electronic correlations.
