This paper presents single lepton and dilepton kinematic distributions measured in dileptonic $t\bar{t}$ events produced in 20.2 $\hbox {fb}^{-1}$ of $\sqrt{s}=8$  TeV pp collisions recorded by the ATLAS experiment at the LHC. Both absolute and normalised differential cross-sections are measured, using events with an opposite-charge $e\mu$ pair and one or two b-tagged jets. The cross-sections are measured in a fiducial region corresponding to the detector acceptance for leptons, and are compared to the predictions from a variety of Monte Carlo event generators, as well as fixed-order QCD calculations, exploring the sensitivity of the cross-sections to the gluon parton distribution function. Some of the distributions are also sensitive to the top quark pole mass, a combined fit of NLO fixed-order predictions to all the measured distributions yields a top quark mass value of ${m_t^{\mathrm {pole}}}=173.2\pm 0.9\pm 0.8\pm 1.2$  GeV, where the three uncertainties arise from data statistics, experimental systematics, and theoretical sources.