  Extreme mass-ratio inspirals~(EMRIs) detectable by the Laser Inteferometric
Space Antenna~(LISA) are unique probes of astrophysics and fundamental physics.
Parameter estimation for these sources is challenging, especially because the
waveforms are long, complicated, known only numerically, and slow to compute in
the most relevant regime, where the dynamics is relativistic. We perform a
time-consuming Fisher-matrix error analysis of the EMRI parameters using
fully-relativistic numerical waveforms to leading order in an adiabatic
expansion on a Kerr background, taking into account the motion of the LISA
constellation, higher harmonics, and also including the leading correction from
the spin of the secondary in the post-adiabatic approximation. We pay
particular attention to the convergence of the numerical derivatives in the
Fisher matrix and to the numerical stability of the covariance matrix, which
for some systems requires computing the numerical waveforms with approximately
$90$-digit precision. Our analysis confirms previous results (obtained with
approximated but much more computationally efficient waveforms) for the
measurement errors on the binary's parameters. We also show that the inclusion
of higher harmonics improves the errors on the luminosity distance and on the
orbital angular momentum angles by one order and two orders of magnitude,
respectively, which might be useful to identify the environments where EMRIs
live. We particularly focus on the measurability of the spin of the secondary,
confirming that it cannot be measured with sufficient accuracy. However, due to
correlations, its inclusion in the waveform model can deteriorate the accuracy
on the measurements of other parameters by orders of magnitude, unless a
physically-motivated prior on the secondary spin is imposed.
