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Analysis of the London penetration depth in Ni-doped <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>CaKFe</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msub><mml:mi>As</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math>
We report combined experimental and theoretical analysis of superconductivity in $\mathrm{CaK}{({\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Ni}}_{x})}_{4}{\mathrm{As}}_{4}$ (CaK1144) for $x=0$, 0.017, and 0.034. To obtain the superfluid density $\ensuremath{\rho}={\left[1+\mathrm{\ensuremath{\Delta}}{\ensuremath{\lambda}}_{L}(T)/{\ensuremath{\lambda}}_{L}(0)\right]}^{\ensuremath{-}2}$, the temperature dependence of the London penetration depth $\mathrm{\ensuremath{\Delta}}{\ensuremath{\lambda}}_{L}(T)$ was measured by using a tunnel-diode resonator (TDR) and the results agreed with the microwave coplanar resonator (MWR) …