We present FlexLLM, a composable High-Level Synthesis (HLS) library for rapid development of domain-specific LLM accelerators. FlexLLM exposes key architectural degrees of freedom for stage-customized inference, enabling hybrid designs that tailor temporal reuse and spatial dataflow differently for prefill and decode, and provides a comprehensive quantization suite to support accurate low-bit deployment. Using FlexLLM, we build a complete inference system for the Llama-3.2 1B model in under two months with only 1K lines of code. The system includes: (1) a stage-customized accelerator with hardware-efficient quantization (12.68 WikiText-2 PPL) surpassing SpinQuant baseline, and (2) a Hierarchical Memory Transformer (HMT) plug-in for efficient long-context processing. On the AMD U280 FPGA at 16nm, the accelerator achieves 1.29$\times$ end-to-end speedup, 1.64$\times$ higher decode throughput, and 3.14$\times$ better energy efficiency than an NVIDIA A100 GPU (7nm) running BF16 inference; projected results on the V80 FPGA at 7nm reach 4.71$\times$, 6.55$\times$, and 4.13$\times$, respectively. In long-context scenarios, integrating the HMT plug-in reduces prefill latency by 23.23$\times$ and extends the context window by 64$\times$, delivering 1.10$\times$/4.86$\times$ lower end-to-end latency and 5.21$\times$/6.27$\times$ higher energy efficiency on the U280/V80 compared to the A100 baseline. FlexLLM thus bridges algorithmic innovation in LLM inference and high-performance accelerators with minimal manual effort.