  We present a joint strong lensing and stellar dynamical framework for future
time-delay cosmography purposes. Based on a pixelated source reconstruction and
the axisymmetric Jeans equations, we are capable of constraining cosmological
distances and hence the current expansion rate of the Universe ($H_0$) to the
few percent level per lens, when high signal-to-noise integral field unit (IFU)
observations from the next generation of telescopes become available. For
illustrating the power of this method, we mock up IFU stellar kinematic data of
the prominent lens system RXJ1131-1231, given the specifications of the James
Webb Space Telescope. Our analysis shows that the time-delay distance
($D_{\Delta t}$) can be constrained with 3.1% uncertainty at best, if future
IFU stellar kinematics are included in the fit and if the set of candidate
model parameterisations contains the true lens potential. These constraints
would translate to a 3.2% precision measurement on $H_0$ in flat $\Lambda$CDM
cosmology from the single lens RXJ1131-1231, and can be expected to yield an
$H_0$ measure with < 2.0% uncertainty, if similar gains in precision can be
reached for two additional lens systems. Moreover, the angular diameter
distance ($D_\mathrm{d}$) to RXJ1131-1231 can be constrained with 2.4%
precision, providing two distance measurements from a single lens system, which
is extremely powerful to further constrain the matter density
($\Omega_\mathrm{m}$). The measurement accuracy of $D_\mathrm{d}$, however, is
highly sensitive to any systematics in the measurement of the stellar
kinematics. For both distance measurements, we strongly advise to probe a large
set of physically motivated lens potentials in the future, to minimise the
systematic errors associated with the lens mass parameterisation.
