。
After the morphine injection, she was free of pain.
Literal
[topic-は] morphine injection [by-means-of-によって] painless [genitive-の] state [into-に] became.
によって here marks the agent or means of a transformation — '(was made into a painless state) by means of a morphine injection.' This is the formal/written agent-marking particle, more weighty than で. 無痛 is a Sino-Japanese compound: 無 ('without') + 痛 ('pain') — 'painless,' as a noun. The の links it to 状態 ('state'). The phrasing has a clinical, medical-report register, fitting for a hospital narrative.