  This paper presents single lepton and dilepton kinematic distributions
measured in dileptonic $t\bar{t}$ events produced in 20.2 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 $m_t^{\rm
pole}=173.2\pm 0.9\pm0.8\pm1.2$ GeV, where the three uncertainties arise from
data statistics, experimental systematics, and theoretical sources.
