Hospital Quality Improvement (QI) plays a critical role in optimizing healthcare delivery by translating high-level hospital goals into actionable solutions. A critical step of QI is to identify the key modifiable contributing factors, a process we call QI factor discovery, typically through expert-driven semi-structured qualitative tools like fishbone diagrams, chart reviews, and Lean Healthcare methods. AI has the potential to transform and accelerate QI factor discovery, which is traditionally time- and resource-intensive and limited in reproducibility and auditability. Nevertheless, current AI alignment methods assume the task is well-defined, whereas QI factor discovery is an exploratory, fuzzy, and iterative sense-making process that relies on complex implicit expert judgments. To design an AI pipeline that formalizes the QI process while preserving its exploratory components, we propose viewing the task as learning not only LLM prompts but also the overarching natural-language specifications. In particular, we map QI factor discovery to steps of the classical AI/ML development process (problem formalization, model learning, and model validation) where the specifications are tunable hyperparameters. Domain experts and AI agents iteratively refine both the overarching specifications and AI pipeline until AI extractions are concordant with expert annotations and aligned with clinical objectives. We applied this "Human-AI Spec-Solution Co-optimization" framework at an urban safety-net hospital to identify factors driving prolonged length of stay and unplanned 30-day readmissions. The resulting AI-for-QI pipelines achieved $\ge 70\%$ concordance with expert annotations. Compared to prior manual Lean analyses, the AI pipeline was substantially more efficient, recovered previous findings, surfaced new modifiable factors, and produced auditable reasoning traces.