Alzheimer's Disease (AD) has become an increasingly critical global health concern, which necessitates effective monitoring solutions in smart health applications. However, the development of such solutions is significantly hindered by the scarcity of AD-specific activity datasets. To address this challenge, we propose SHADE-AD, a Large Language Model (LLM) framework for Synthesizing Human Activity Datasets Embedded with AD features. Leveraging both public datasets and our own collected data from 99 AD patients, SHADE-AD synthesizes human activity videos that specifically represent AD-related behaviors. By employing a three-stage training mechanism, it broadens the range of activities beyond those collected from limited deployment settings. We conducted comprehensive evaluations of the generated dataset, demonstrating significant improvements in downstream tasks such as Human Activity Recognition (HAR) detection, with enhancements of up to 79.69%. Detailed motion metrics between real and synthetic data show strong alignment, validating the realism and utility of the synthesized dataset. These results underscore SHADE-AD's potential to advance smart health applications by providing a cost-effective, privacy-preserving solution for AD monitoring.