Supporting older adults in health self-management is crucial for promoting independent aging, particularly given the growing strain on healthcare systems. While voice assistants (VAs) hold the potential to support aging in place, they often lack tailored assistance and present usability challenges. We addressed these issues through a five-stage design process with older adults to develop a personal health assistant. Starting with in-home interviews (N = 17), we identified two primary challenges in older adult's health self-management: health awareness and medical adherence. To address these challenges, we developed a high-fidelity LLM-powered VA prototype to debrief doctor's after-visit summary and generate tailored medication reminders. We refined our prototype with feedback from co-design workshops (N = 10) and validated its usability through in-home studies (N = 5). Our work highlights key design features for personal health assistants and provides broader insights into desirable VA characteristics, including personalization, adapting to user context, and respect for user autonomy.