Pretrained large language models (LLMs) have excelled in a variety of natural language processing (NLP) tasks, including summarization, question answering, and translation. However, LLMs pose significant security risks due to their tendency to memorize training data, leading to potential privacy breaches and copyright infringement. Therefore, accurate measurement of the memorization is essential to evaluate and mitigate these potential risks. However, previous attempts to characterize memorization are constrained by either using prefixes only or by prepending a constant soft prompt to the prefixes, which cannot react to changes in input. To address this challenge, we propose a novel method for estimating LLM memorization using dynamic, prefix-dependent soft prompts. Our approach involves training a transformer-based generator to produce soft prompts that adapt to changes in input, thereby enabling more accurate extraction of memorized data. Our method not only addresses the limitations of previous methods but also demonstrates superior performance in diverse experimental settings compared to state-of-the-art techniques. In particular, our method can achieve the maximum relative improvement of 135.3% and 39.8% over the vanilla baseline on average in terms of *discoverable memorization rate* for the text generation task and code generation task, respectively. Our code is available at https://github.com/wangger/llm-memorization-dsp.