  Time-delay strong lensing (TDSL) is a powerful probe of the current expansion
rate of the Universe. However, in light of the discrepancies between early and
late-time cosmological studies, efforts revolve around the characterisation of
systematic uncertainties in the methods. Here, we focus on the mass-sheet
degeneracy (MSD), which is considered a significant source of systematics in
TDSL, and aim to assess the constraining power provided by IFU stellar
kinematics. We approximate the MSD with a cored, two-parameter extension to the
lensing mass profiles (with core radius $r_{\rm c}$ and mass-sheet parameter
$\lambda_{\rm int}$). In addition, we utilise mock IFU stellar kinematics of
time-delay strong lenses, given the prospects of obtaining such data with JWST.
We construct joint strong lensing and stellar dynamical models, where the time
delays, mock imaging and IFU observations are used to constrain the mass
profile of lens galaxies, and yield joint constraints on the time-delay
distance ($D_{\Delta t}$) and angular diameter distance ($D_{\rm d}$) to the
lens. We find that mock JWST-like stellar kinematics constrain the internal
mass sheet and limit its contribution to the uncertainties of $D_{\Delta t}$
and $D_{\rm d}$, each at the < 4% level, without assumptions on the background
cosmological model. These distance constraints would translate to a < 4%
precision measurement on $H_{\rm 0}$ in flat $\Lambda CDM$ for a single lens.
Our study shows that IFU stellar kinematics of time-delay strong lenses will be
key in lifting the MSD on a per lens basis, assuming reasonable core sizes.
However, even in the limit of infinite $r_{\rm c}$, where $D_{\Delta t}$ is
degenerate with $\lambda_{\rm int}$, stellar kinematics of the deflector, time
delays and imaging data will provide powerful constraints on $D_{\rm d}$, which
becomes the dominant source of information in the cosmological inference.
