You can think of it as two clauses connected with "and". 1) 잠도 안 자다: "잠을 안 자다" is "not sleep". "잠도" makes it into "not even sleep" (잠을 자다 = "sleep"; 안 = negation adverb; 도 = as well; even). 2) 영어 공부 하고 있다: literally "I am doing English study", so "I am studying English". "-고 있다" is the present progressive tense, like "-ing" form in English. "-고" of "안 자고" is like "and", a very common form for sequencing two or more clauses into one sentence making a loose connection in meaning. So 잠도 안 자고 영어 공부 하고 있어요 is literally "I am not sleeping AND I am studying English", which is like "I am staying up late studying [to study] English". This -고 (sometimes -고서 to emphasize the sequence) is one of the most heavily used connective form. You'll see it all over the place all the time: - 점심을 먹고(서) 밖에 나가다. - 길을 못 찾고 헤맸다. - 눈물을 머금고 후퇴했다. - 수술을 하고(서) 경과를 볼 것이다.