  In this series of papers, we investigate the spreading and vanishing dynamics
of time almost periodic diffusive KPP equations with free boundaries. Such
equations are used to characterize the spreading of a new species in time
almost periodic environments with free boundaries representing the spreading
fronts. In the first part of the series, we showed that a spreading-vanishing
dichotomy occurs for such free boundary problems (see [16]). In this second
part of the series, we investigate the spreading speeds of such free boundary
problems in the case that the spreading occurs. We first prove the existence of
a unique time almost periodic semi-wave solution associated to such a free
boundary problem. Using the semi-wave solution, we then prove that the free
boundary problem has a unique spreading speed.
