Equilibrium and nonequilibrium free energies of complex fluids are fundamental quantities that can be used to determine a wide array of system properties. Recently, we demonstrated the direct determination of the equilibrium free energy landscape and corresponding elasticity of polymer chains from work calculations in highly nonequilibrium fluid flows.(1) In the present study, we further demonstrate the generality of this formalism by applying this method to polymeric systems driven by fluid flows with vorticity and for molecules with dominant intramolecular hydrodynamic interactions (HI). We employ Brownian dynamics simulations of double stranded DNA with fluctuating HI, and we analyze polymer dynamics and the resultant free energy calculations in the context of the nonequilibrium work relations. Furthermore, we investigate the role of HI on the work and housekeeping power required to maintain a polymer chain at a nonequilibrium steady-state in flow, and we consider the relationship between housekeeping power and polymer chain size. On the basis of the results in this study, nonequilibrium work relations appear to be a powerful set of tools that can be used to understand the behavior of polymeric systems and soft materials.