AI inference clusters are increasingly constrained by instantaneous power, not just energy: grid operators condition new capacity on demand response, imposing time-varying power caps. Existing LLM serving systems optimize a static energy objective or shed fixed priority tiers under load; either way, goodput collapses when the power envelope moves. An LLM pipeline is not a uniform load: compute-bound prefill loses throughput almost linearly with GPU frequency, memory-bound answer decode sustains it down to 0.57x nominal, and reasoning's thinking phase couples KV-cache capacity to scheduling – so a cap should be steered to where each watt costs the least performance. PowerSlider does so with a new Flex SLO contract that turns bounded user slack into an optimization constraint, prefill–think–answer disaggregation exposing per-stage frequency and KV control, and a Karush–Kuhn–Tucker (KKT) online solver re-solving within 7.7 ms of every cap change, backed by a consolidated fail-safe that power-gates drained instances when DVFS bottoms out on static power. On SGLang with production traces, PowerSlider sustains 78.3% online goodput at a 30% cap reduction versus 47.6% for the best of five baselines (1.64x), holds latency-critical tails within 1.3x of nominal (baselines: 2.3–6x, up to 12x), and delivers 92% mean goodput through a replayed CAISO grid-emergency day bottoming at 0.41x (54% at the trough; every baseline below 7%).