We present BioSpark, a system for analogical innovation designed to act as a creativity partner in reducing the cognitive effort in finding, mapping, and creatively adapting diverse inspirations. While prior approaches have focused on initial stages of finding inspirations, BioSpark uses LLMs embedded in a familiar, visual, Pinterest-like interface to go beyond inspiration to supporting users in identifying the key solution mechanisms, transferring them to the problem domain, considering tradeoffs, and elaborating on details and characteristics. To accomplish this BioSpark introduces several novel contributions, including a tree-of-life enabled approach for generating relevant and diverse inspirations, as well as AI-powered cards including 'Sparks' for analogical transfer; 'Trade-offs' for considering pros and cons; and 'Q&A' for deeper elaboration. We evaluated BioSpark through workshops with professional designers and a controlled user study, finding that using BioSpark led to a greater number of generated ideas; those ideas being rated higher in creative quality; and more diversity in terms of biological inspirations used than a control condition. Our results suggest new avenues for creativity support tools embedding AI in familiar interaction paradigms for designer workflows.