We report on progress towards calculating the NNLO cross section for top quark pair production in the the framework which combines the $q_T$-slicing method and small-$q_T$ factorization. In this contribution, we present the complete result for the small-$q_T$ soft function at NNLO, which constitutes the main challenge on the way to our goal. In order to evaluate the divergent integrals appearing in the calculation, we develop methods based on sector decomposition and differential equations. We present complete validation of our framework and provide results which are ready for application in the calculation of the $t\bar t$ cross section at NNLO.