。
She spoke to the chairperson.
Literal
She [topic-は] chairperson [to-に] addressed.
話しかける is a compound verb (話す + かける) meaning 'to speak to, to address.' The かける suffix here is one of its many uses: 'to direct an action toward someone' — see also 呼びかける ('call out to'), 笑いかける ('smile at'), 手をかける ('lay a hand on'). The に on 議長 marks the recipient of the addressing. 議長 itself is the formal 'chairperson, head of a meeting,' built from 議 ('discussion, deliberation') + 長 ('chief, head').