The results of extensive all-atom molecular dynamics (MD) simulations of water- and methanol-solvated SPEEK (sulfonated poly(ether ether ketone)) are reported. In this Part II of the two-part article, we present results elucidating the spatial distributions of hydronium ion (vehicular proton) and methanol and the transport properties of water, hydronium ions, and methanol. Our results suggest that hydronium ions escape attraction shells of sulfonic groups with increasing water and methanol contents but move closer to sulfur with an increase in temperature. The localization of the hydronium ion near sulfonate anion was seen to be significantly more pronounced than in Nation, suggesting stronger basicity of sulfonate anion and therefore weaker acidity of its conjugate acid in SPEEK than in Nafion. In contrast with Nation, methanol competes with hydronium ions and water to solvate sulfonate anion and also lies closer to aromatic backbone. Water diffusion coefficients follow experimentally observed trends where they are lower in SPEEK than in Nation at low water weight percent but approach the Nation values at higher water weight percent. The vehicular proton diffusivity, as quantified by hydronium ion diffusivity, was found to be an order of magnitude lower than that in Nafion. The transport results are rationalized based on the structural insights presented in Part I and the present article.