  We report new Northern Extended Millimeter Array (NOEMA) observations of the
[CII], [NII] and [OI] atomic fine structure lines and dust continuum emission
of J1148+5251, a z=6.42 quasar, that probe the physical properties of its
interstellar medium (ISM). The radially-averaged [CII] and dust continuum
emission have similar extensions (up to $\theta = 2.51^{+0.46}_{-0.25}\
\rm{arcsec}$, corresponding to $r= 9.8^{+3.3}_{-2.1}\ \rm{kpc}$ accounting for
beam-convolution), confirming that J1148+5251 is the quasar with the largest
[CII]-emitting has reservoir known at these epochs.Moreover, if the [CII]
emission is examined only along its NE-SW axis, a significant excess
($>5.8\sigma$) of [CII] emission (with respect to the dust) is detected. The
new wide--bandwidth observations enable us to accurately constrain the
continuum emission, and do not statistically require the presence of broad
[CII] line wings that were reported in previous studies. We also report the
first detection of the [OI] and (tentatively) [NII] emission lines in
J1148+5251. Using Fine Structure Lines (FSL) ratios of the [CII], [NII], [OI]
and previously measured [CI] emission lines, we show that J1148+5251 has
similar ISM conditions compared to lower--redshift (ultra)-luminous infrared
galaxies. CLOUDY modelling of the FSL ratios exclude X--ray dominated regions
(XDR) and favours photodissociation regions (PDR) as the origin of the FSL
emission. We find that a high radiation field ($10^{3.5-4.5}\,G_0$), high gas
density ($n \simeq 10^{3.5-4.5}\, \rm{cm}^{-3}$) and HI column density of
$10^{23} \,\rm{cm^{-2}}$ reproduce the observed FSL ratios well.
