  Deep near-infrared photometric surveys are efficient in identifying
high-redshift galaxies, however they can be prone to systematic errors in
photometric redshift. This is particularly salient when there is limited
sampling of key spectral features of a galaxy's spectral energy distribution
(SED), such as for quiescent galaxies where the expected age-sensitive
Balmer/4000 A break enter the $K$-band at $z>4$. With single filter sampling of
this spectral feature, degeneracies between SED models and redshift emerge. A
potential solution to this comes from splitting the $K$-band into multiple
filters. We use simulations to show an optimal solution is to add two
medium-band filters, $K_\mathrm{blue}$ ($\lambda_\mathrm{cen}$=2.06 $\mu$m,
$\Delta\lambda$=0.25 $\mu$m) and $K_\mathrm{red}$ ($\lambda_\mathrm{cen}$=2.31
$\mu$m, $\Delta\lambda$=0.27 $\mu$m), that are complementary to the existing
$K_\mathrm{s}$ filter. We test the impact of the $K$-band filters with
simulated catalogues comprised of galaxies with varying ages and
signal-to-noise. The results suggest that the $K$-band filters do improve
photometric redshift constraints on $z>4$ quiescent galaxies, increasing
precision and reducing outliers by up to 90$\%$. We find that the impact from
the $K$-band filters depends on the signal-to-noise, the redshift and the SED
of the galaxy. The filters we designed were built and used to conduct a pilot
of the FLAMINGOS-2 Extra-galactic Near-Infrared $K$-band Split (FENIKS) survey.
While no new $z>4$ quiescent galaxies are identified in the limited area pilot,
the $K_\mathrm{blue}$ and $K_\mathrm{red}$ filters indicate strong Balmer/4000
A breaks in existing candidates. Additionally we identify galaxies with strong
nebular emission lines, for which the $K$-band filters increase photometric
redshift precision and in some cases indicate extreme star-formation.
