We sat down and talked through the whole thing until we reached agreement.
Literal
We [topic-は] agreement [to-に] reach until sit-and one-part-start-end talk-together.
同意に達する ('reach agreement') uses に for the thing reached. 一部始終 is a compact four-character compound (四字熟語-adjacent) meaning 'from start to finish, the whole thing' — from 一部 (one part) 始 (start) 終 (end). 話し合う is a reciprocal compound verb ('talk together, discuss mutually'). The chained verbs create a vivid scene of sustained discussion.