Pre-training on public data is an effective method to improve the performance for federated learning (FL) with differential privacy (DP). This paper investigates how large language models (LLMs) trained on public data can improve the quality of pre-training data for the on-device language models trained with DP and FL. We carefully design LLM prompts to filter and transform existing public data, and generate new data to resemble the real user data distribution. The model pre-trained on our synthetic dataset achieves relative improvement of 19.0% and 22.8% in next word prediction accuracy compared to the baseline model pre-trained on a standard public dataset, when evaluated over the real user data in Gboard (Google Keyboard, a production mobile keyboard application). Furthermore, our method achieves evaluation accuracy better than or comparable to the baseline during the DP FL fine-tuning over millions of mobile devices, and our final model outperforms the baseline in production A/B testing. Our experiments demonstrate the strengths of LLMs in synthesizing data close to the private distribution even without accessing the private data, and also suggest future research directions to further reduce the distribution gap.