News media often leverage documents to find ideas for stories, while being critical of the frames and narratives present. Developing angles from a document such as a press release is a cognitively taxing process, in which journalists critically examine the implicit meaning of its claims. Informed by interviews with journalists, we developed AngleKindling, an interactive tool which employs the common sense reasoning of large language models to help journalists explore angles for reporting on a press release. In a study with 12 professional journalists, we show that participants found AngleKindling significantly more helpful and less mentally demanding to use for brainstorming ideas, compared to a prior journalistic angle ideation tool. AngleKindling helped journalists deeply engage with the press release and recognize angles that were useful for multiple types of stories. From our findings, we discuss how to help journalists customize and identify promising angles, and extending AngleKindling to other knowledge-work domains.