The rising prevalence of chronic wounds, especially in aging populations, presents a significant healthcare challenge due to prolonged hospitalizations, elevated costs, and reduced patient quality of life. Traditional wound care is resource-intensive, requiring frequent in-person visits that strain both patients and healthcare professionals (HCPs). Therefore, we present WoundAIssist, a patient-centered, AI-driven mobile application designed to support telemedical wound care. WoundAIssist enables patients to regularly document wounds at home via photographs and questionnaires, while physicians remain actively engaged in the care process through remote monitoring and video consultations. A distinguishing feature is an integrated lightweight deep learning model for on-device wound segmentation, which, combined with patient-reported data, enables continuous monitoring of wound healing progression. Developed through an iterative, user-centered process involving both patients and domain experts, WoundAIssist prioritizes an user-friendly design, particularly for elderly patients. A conclusive usability study with patients and dermatologists reported excellent usability, good app quality, and favorable perceptions of the AI-driven wound recognition. Our main contribution is two-fold: (I) the implementation and (II) evaluation of WoundAIssist, an easy-to-use yet comprehensive telehealth solution designed to bridge the gap between patients and HCPs. Additionally, we synthesize design insights for remote patient monitoring apps, derived from over three years of interdisciplinary research, that may inform the development of similar digital health tools across clinical domains.