The goal of kidney exchange programs is to match recipients with a willing but incompatible donor with another compatible donor, so as to maximize total (weighted) transplants. There is significant uncertainty in this process, as planned transplants may be cancelled for a variety of reasons. Planning exchanges while considering failures, and options for recourse, is therefore crucial. In this paper, we reconsider a robust optimization model with recourse proposed in Carvalho et al. (2020) that takes into account the event that a number of donors leaves the KEP. After these donors have left the program, a new set of exchanges is identified (the recourse decision). Current algorithmic considerations do not allow to find optimal solutions for the robust optimization model for realistic sized kidney exchanges with a reasonable time frame. Following the iterative framework in Carvalho et al. (2020) for solving a large-scale mixed integer programming model, we propose a new variable and constraint generation method based on Generalized Benders' Decomposition. A characterization of this method is given based on two widely used integer programming models for kidney exchange programs. Furthermore, a lifting technique is proposed to obtain stronger Benders cuts to speed up computation. Computational results show that our algorithm is very competitive, improving on the running time of the state-of-the-art method by one order of magnitude. Furthermore, our methods are able to solve a large number of previously unsolved instances within the same time limit.