We present a search for the lepton flavor violating decays B^{+}→K^{+}τ^{±}ℓ^{∓}, with ℓ=(e,μ), using the full data sample of 772×10^{6} BB[over ¯] pairs recorded by the Belle detector at the KEKB asymmetric-energy e^{+}e^{-} collider. We use events in which one B meson is fully reconstructed in a hadronic decay mode. We find no evidence for B^{±}→K^{±}τℓ decays and set upper limits on their branching fractions at the 90% confidence level in the (1-3)×10^{-5} range. The obtained limits are the world's best results.