Advancements in cooling for applications such as gas turbines components require improved understanding of the complex heat transfer mechanisms and the interactions between those mechanisms. Critical cooling applications often rely on multiple thermal protection techniques, including internal cooling and external film cooling in gas turbine airfoils, to efficiently cool components and limit the use of coolant. Most research to quantify the effectiveness of such cooling technologies for gas turbine applications has isolated internal and external cooling in separate experiments. The research presented in this paper uses a conjugate heat transfer approach to account for the combined effects of both internal and external cooling. The geometry used for this study is a turbine blade endwall that includes impingement and film cooling as well as the relevant conduction through the endwall. Appropriate geometric and flow parameters were scaled to ensure engine relevant dimensionless temperatures were obtained. Using the conjugate heat transfer approach, the effect of varying the height of the impingement channel was examined using spatially resolved external wall temperatures obtained from both experiments and simulations. A one-dimensional heat transfer analysis was used to derive the average internal heat transfer coefficients from the experimental results. Both experiments and simulations showed good agreement between area averaged cooling effectiveness and impingement heat transfer coefficients. The cooling effectiveness and heat transfer coefficients peaked for an impingement channel height of around three impingement hole diameters. However, the heat transfer coefficients were more sensitive than the overall effectiveness to the changes in height of the impingement channel.