We seek to determine whether a small number of measurements of upper ocean temperature and currents can be used to make estimates of the drag coefficient that have a smaller range of uncertainty than previously found. We adopt a numerical approach in an inverse problem setup using an ocean model and its adjoint, to assimilate data and to adjust the drag coefficient parameterization (here the free parameter) with wind speed that corresponds to the minimum of a model minus data misfit or cost function. Pseudo data are generated from a reference forward simulation, and are perturbed with different levels of Gaussian distributed noise. It is found that it is necessary to assimilate both surface current speed and temperature data to obtain improvement over previous estimates of the drag coefficient. When data is assimilated without any smoothing or constraints on the solution, the drag coefficient is overestimated at low wind speeds and there are unrealistic, high frequency oscillations in the adjusted drag coefficient curve. When second derivatives of the drag coefficient curve are penalized and the solution is constrained to experimental values at low wind speeds, the adjusted drag coefficient is within 10% of its target value. This result is robust to the addition of realistic random noise meant to represent turbulence due to the presence of mesoscale background features in the assimilated data, or to the wind speed time series to model its unsteady and gusty character. When an eddy is added to the background flow field in both the initial condition and the assimilated data time series, the target and adjusted drag coefficient are within 10% of one another, regardless of whether random noise is added to the assimilated data. However, when the eddy is present in the assimilated data but is not in the initial conditions, the drag coefficient is overestimated by as much as 30%. This carries the implication that when real data is assimilated, care needs to be taken in representation of the initial state of the ocean, and especially the field of background currents.