Measurements of the real 蟽1 and imaginary 蟽2 part of the conductivity were performed in optimally doped BaFe1.9Ni0.1As2 and overdoped BaFe1.88Ni0.12As2 crystals in the frequency range 20 MHz-1.5 GHz using a single coil technique. The temperature dependence of the London penetration depth follows a T2 law. The conductivity 蟽1 increases with decreasing temperature below Tc in agreement with the results obtained for the optimally Co doped BaFe2鈭?em>xCoxAs2 crystals. The increase of 蟽1 in the superconducting state is attributed to a rapidly decrease of the quasiparticle scattering rate.