Kelvin-Helmholtz instability (KHI) in solid media plays an important role in fields such as inertial confinement fusion (ICF) and high-energy-density physics (HEDP). While KHI in fluids has been extensively investigated, its manifestation in solids remains far less understood. Recent theoretical studies have begun to address KHI at interfaces between an ideal fluid and an elastic–perfectly plastic solid; however, these models are generally restricted to idealized interfaces with discontinuous jumps in velocity and density. In many realistic scenarios, including inclined shock loading of solid interfaces, high-velocity impact welding, and ICF target compression, velocity and density instead vary continuously and asymmetrically across finite interfacial layers. To address this limitation, we develop a theoretical model for KHI at an interface between an ideal fluid and an elastic–perfectly plastic solid that explicitly incorporates continuous velocity and density gradients. We consider an ideal fluid flowing past a metal surface perturbed by a small-amplitude sinusoidal disturbance with wavelength λ (wavenumber k = 2π/λ) and amplitude ξ₀. The perturbed interface separates a solid, characterized by shear modulus G_s, yield strength Y_s, and far-field density ρ_s,0, from a fluid with far-field density ρ_f,0. Both velocity and density vary continuously across characteristic thicknesses L_u and L_ρ. The perturbation flow is assumed irrotational and described using velocity potentials, while the elastic–plastic response of the solid is incorporated through interfacial normal–stress conditions. By introducing dimensionless parameters—the velocity and density gradient parameters α = 1/(1+kL_u) and β = 1/(1+kL_ρ), the normalized wavenumber \hat\lambda=2 \pi \xi_0 / \lambda and the normalized strength \hatY=\rho_s, 0 U_f, 0^2 / Y_s—we derive analytical expressions for the elastic–plastic transition (EP), the instability boundary (IB), and the temporal evolution of the perturbation amplitude. A decrease in the velocity gradient parameter α stabilizes the interface, converting sustained growth of the perturbation into small-amplitude oscillations after strength suppression. In the (\hat\lambda, \hatY) parameter space, this stabilization manifests as an expansion of the stable region, with both the EP and IB curves shifting away from the coordinate axes. The effect of the density gradient depends on the Atwood number A_t. For A_t > 0, a decrease in β destabilizes the interface, shifting EP and IB curves toward the axes and shrinking the stable region; for A_t < 0, a decrease in β instead promotes stability. When both velocity and density gradients are present and A_t > 0, their competing effects are governed by the ratio T = β / α: for sufficiently large T, the net effect is destabilizing, whereas for sufficiently small T, the net effect is stabilized.