Cognitive health in older adults presents a growing challenge. Although conversational interventions show feasibility in improving cognitive wellness, human caregiver resources remain overloaded. AI-based chatbots have shown promise, yet existing work is often limited to implicit strategies or heavily depends on training and label resources. In response, we propose a strategy-guided AI chatbot named ChatWise that follows a dual-level conversation reasoning framework. It integrates macro-level strategy planning and micro-level utterance generation to enable engaging, multi-turn dialogue tailored to older adults. Empirical results show that ChatWise closely aligns with professional human caregiver behaviors in offline evaluation using real clinic data, and achieves positive user cognitive and emotional responses in interactive simulations with digital twins, which significantly outperforms AI baselines that follow implicit conversation generation.