Large Language Model (LLM) inference relies on key-value (KV) caches to avoid redundant attention computation. While approximate KV cache retention techniques reduce memory usage by sacrificing model accuracy, lossless approaches instead evict KV cache blocks from GPU memory and reconstruct them on demand to preserve exact outputs. Existing lossless KV cache management systems primarily base eviction decisions on access frequency or positional heuristics, without considering how different KV cache blocks affect the execution efficiency of GPU attention kernels. In this paper, we propose AsymCache, a computation-latency-aware KV cache management system for LLM inference that explicitly aligns cache residency decisions with GPU attention kernel performance, including three key components: Multi-Segment Attention (MSA) for efficient non-contiguous KV context processing, a cache eviction policy that jointly optimizes hit rate and position-aware recomputation cost, and an adaptive chunking scheduler for high hardware utilization. Experiments show that AsymCache reduces TTFT by up to 1.90-2.03x and time-per-output-token (TPOT) by 1.62-1.71x over latest baselines, confirming the effectiveness of the method in common workloads and validating its design goal of balancing computational efficiency with cache hit rate. Moreover, the low-level design of AsymCache allows seamless integration into agent serving systems such as Continuum, where it further reduces average job latency by up to 18.1%.