Molecular editing-modifying a given molecule to improve desired properties-is a fundamental task in drug discovery. While LLMs hold the potential to solve this task using natural language to drive the editing, straightforward prompting achieves limited accuracy. In this work, we propose AgentDrug, an agentic workflow that leverages LLMs in a structured refinement process to achieve significantly higher accuracy. AgentDrug defines a nested refinement loop: the inner loop uses feedback from cheminformatics toolkits to validate molecular structures, while the outer loop guides the LLM with generic feedback and a gradient-based objective to steer the molecule toward property improvement. We evaluate AgentDrug on benchmarks with both single- and multi-property editing under loose and strict thresholds. Results demonstrate significant performance gains over previous methods. With Qwen-2.5-3B, AgentDrug improves accuracy by 20.7% (loose) and 16.8% (strict) on six single-property tasks, and by 7.0% and 5.3% on eight multi-property tasks. With larger model Qwen-2.5-7B, AgentDrug further improves accuracy on 6 single-property objectives by 28.9% (loose) and 29.0% (strict), and on 8 multi-property objectives by 14.9% (loose) and 13.2% (strict).