The development of a visco-resistive length scale for the thickness of a reconnecting current sheet would have significant consequences for the physics of magnetic reconnection in solar flares. In this paper, planar magnetic reconnection in an incompressible visco-resistive plasma is investigated analytically and numerically. Relaxation simulations are performed in an "open" geometry that allows material to enter and exit the reconnection volume. Solutions of two types are identified depending on the strength of the external flow that drives the reconnection. For sufficiently strong flows separate resistive and viscous layers develop in the reconnection region. In this case merging rates are found to be largely independent of viscosity. However, when the flow is too weak to produce a localized current layer, an equilibrium lacking any small-scale structure is obtained. The central conclusion is that neither of these steady-state solutions provide evidence of a visco-resistive length scale.