ㄴ가 해내다

pull something off, achieve something

Every line here is 뭔가 "something" in front of 해내다, "pull off, manage to do". 해내다 is 하다 with the sense of getting through to a result, so it is achievement rather than activity: I thought I could make something happen even after failing four times, the satisfaction of having achieved something, wanting to accomplish something from here on. The 뭔가 keeps the goal unnamed, which is why these lines are about drive rather than about any specific task.