  We investigate the $K_1(1270)-K_1(1400)$ mixing caused by the flavor $SU(3)$
symmetry breaking. The mixing angle is expressed by a $K_{1A}\to K_{1B}$ matrix
element induced by the operators that breaks flavor $SU(3)$ symmetry. The QCD
contribution to this matrix element is assumed to be dominated and calculated
by QCD sum rules. A three-point correlation function is defined and handled
both at the hadron and quark-gluon levels. The quark-gluon level calculation is
based on operator product expansion up to dimension-5 condensates. A detailed
numerical analysis is performed to determine the Borel parameters, and the
obtained mixing angle is $\theta_{K_1}=22^{\circ}\pm 7^{\circ}$ or
$\theta_{K_1}=68^{\circ}\pm 7^{\circ}$.
