Reconfigurable antenna systems have gained much attention for potential use in the next generation wireless systems. However, conventional direction-of-arrival (DoA) estimation algorithms for antenna arrays cannot be used directly in reconfigurable antennas due to different design of the antennas. In this paper, we present an adjacent pattern power ratio (APPR) algorithm for two-port composite right/left-handed (CRLH) reconfigurable leaky-wave antennas (LWAs). Additionally, we compare the performances of the APPR algorithm and LWA-based MUSIC algorithms. We study how the computational complexity and the performance of the algorithms depend on number of selected radiation patterns. In addition, we evaluate the performance of the APPR and MUSIC algorithms with numerical simulations as well as with real world indoor measurements having both line-of-sight and non-line-of-sight components. Our performance evaluations show that the DoA estimates are in a considerably good agreement with the real DoAs, especially with the APPR algorithm. In summary, the APPR and MUSIC algorithms for DoA estimation along with the planar and compact LWA layout can be a valuable solution to enhance the performance of the wireless communication in the next generation systems.