Current techniques for privacy auditing of large language models (LLMs) have limited efficacy -- they rely on basic approaches to generate canaries which leads to weak membership inference attacks that in turn give loose lower bounds on the empirical privacy leakage. We develop canaries that are far more effective than those used in prior work under threat models that cover a range of realistic settings. We demonstrate through extensive experiments on multiple families of fine-tuned LLMs that our approach sets a new standard for detection of privacy leakage. For measuring the memorization rate of non-privately trained LLMs, our designed canaries surpass prior approaches. For example, on the Qwen2.5-0.5B model, our designed canaries achieve $49.6\%$ TPR at $1\%$ FPR, vastly surpassing the prior approach's $4.2\%$ TPR at $1\%$ FPR. Our method can be used to provide a privacy audit of $\varepsilon \approx 1$ for a model trained with theoretical $\varepsilon$ of 4. To the best of our knowledge, this is the first time that a privacy audit of LLM training has achieved nontrivial auditing success in the setting where the attacker cannot train shadow models, insert gradient canaries, or access the model at every iteration.