  NGC 6441 is a bulge globular cluster with an unusual horizontal branch
morphology and a rich population of RR Lyrae (RRL) and Type II Cepheid (T2C)
variables that is unexpected for its relatively high metallicity. We present
near-infrared (NIR, $JHK_s$) time-series observations of 42 RRL, 8 T2Cs, and 10
eclipsing binary candidate variables in NGC 6441. The multi-epoch observations
were obtained using the FLAMINGOS-2 instrument on the 8-m Gemini South
telescope. Multi-band data are used to investigate pulsation properties of RRL
and T2C variables including their light curves, instability strip,
period--amplitude diagrams, and period--luminosity and period--wesenheit
relations (PLRs and PWRs) in $JHK_s$ filters. NIR pulsation properties of RRL
variables are well fitted with theoretical models that have canonical helium
content and mean-metallicity of NGC 6441 ([Fe/H]$=-0.44\pm0.07$~dex). The
helium-enhanced RRL models predict brighter NIR magnitudes and bluer colors
than the observations of RRL in the cluster. This suggests that these RRL
variables in NGC 6441 are either not significantly helium enhanced as
previously thought or the impact of such enhancement is smaller in NIR than the
predictions of the pulsation models. We also use theoretical calibrations of
RRL period--luminosity--metallicity (PLZ) relations to simultaneously estimate
the mean reddening, $E(J-K_s)=0.26\pm0.06$~mag, and the distance,
$d=12.67\pm0.09$~kpc, to NGC 6441. Our mean reddening value and the distance
are consistent with the independent estimates from the bulge reddening map
based on red clump stars and the latest Gaia data, respectively. Our distance
and reddening values provide a very good agreement between the PLRs of T2Cs in
NGC 6441 and those for RRL and T2Cs in Galactic globular clusters that span a
broad range of metallicity.
