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 galaxys 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, $Deltalambda$=0.25 $mu$m) and $K_mathrm{red}$ ($lambda_mathrm{cen}$=2.31 $mu$m, $Deltalambda$=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.
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