Kidney exchange has been studied extensively from the perspective of market design, and a significant focus has been on better algorithms for finding chains and cycles to increase the number of possible matches. A more dramatic benefit could come from incorporating compatible pairs into the mechanism, but this possibility has been relatively understudied. In order to incentivize a compatible pair to participate in exchange, they must be offered a higher quality match for the recipient that can be performed without adding extra waiting time. In this paper, we make two main contributions to the study of incorporating compatible pairs in exchanges. First, we leverage the recently proposed Living Donor Kidney Profile Index (LKDPI) to measure match quality, and develop a novel simulator (based on data from a major transplant center) for the joint distribution of compatibility and quality across pairs. This simulator allows us to study the benefits of including compatible pairs under different models and assumptions. Second, we introduce a hybrid online/batch matching model with impatient (compatible) and patient (incompatible) pairs to capture the need for immediacy. We introduce new algorithms for matching in this model, including one based on online primal-dual techniques. Overall, our results indicate great potential in terms of both increased numbers of transplants of incompatible pairs (almost doubling the number transplanted) as well as improved match quality for recipients in compatible pairs (increasing expected graft survival by between 1 and 2 years). The results are also promising for hard-to-match subpopulations, including blood group O recipients.