The electromagnetic form factors of charged and neutral kaons are strongly constrained by their low-energy singularities, in the isovector part from two-pion intermediate states and in the isoscalar contribution in terms of ω and ϕ residues. The former can be predicted using the respective ππ → K^-K partial-wave amplitude and the pion electromagnetic form factor, while the latter parameters need to be determined from electromagnetic reactions involving kaons. We present a global analysis of time- and spacelike data that implements all of these constraints. The results enable manifold applications: kaon charge radii, elastic contributions to the kaon electromagnetic self energies and corrections to Dashen's theorem, kaon boxes in hadronic light-by-light (HLbL) scattering, and the ϕ region in hadronic vacuum polarization (HVP). Our main results are: ⟨r²⟩_c = 0.359(3) fm², ⟨r²⟩_n = -0.060(4) fm² for the charged and neutral radii, ϵ = 0.63(40) for the elastic contribution to the violation of Dashen's theorem, a_μ^K-box = -0.48(1) × 10^-11 for the charged kaon box in HLbL scattering, and a_μ^HVP[K⁺K^-, ≤ 1.05 GeV] = 184.5(2.0) × 10^-11, a_μ^HVP[K_SK_L, ≤ 1.05 GeV] = 118.3(1.5) × 10^-11 for the HVP integrals around the ϕ resonance. The global fit to K^-K gives M^-_ϕ = 1019.479(5) MeV, Γ^-_ϕ = 4.207(8) MeV for the ϕ resonance parameters including vacuum-polarization effects.