Not all ending ㄴs are the same.
끝장낸 is a present perfect tense (completion) of the verb 끝장내다 in an adjective form.
다음엔 is short for 다음에는. So the two ending ㄴs are completely different things.
-ㄴ 다음에(는) is a common structure for "after V-ing" (V representing an action), similar to 끝장내고 그 다음에는.
다음에 and 다음에는 are about the same, but without 는 it sounds more dry and factual, while with it, it's more colloquial and rhythmical.
The phrase means "after finishing the drink(음료) ...", where 끝장낸 (having taken care of something) is used humorously (끝장내다 is normally serious, like to see the end of something, or to put and end).
As a similar phrase: 영화를 본 다음에/다음엔 뭐 할까? = What should we do after watching the movie?