"잠도 안 자고 영어 공부 하고 있어요." "I’m even trying to stay awake and I’m studying English." doesn't seem to be a good translation. It is trying to capture the sense of 도 (as well, even) as "even trying to", but the overall phrase doesn't sound very natural. We typically say 잠을 자다 or its short form 잠자다 for "to sleep", although sometimes they're shortened to just 자다. The negative of it is 잠(을) 안 자다, or its long form 잠을 자지 않다. 잠을 안 자다 is a rather neutral sounding phrase though. If you want to emphasize it like "doesn't even sleep" or "burns the midnight oil", etc, we use 도, like 잠도 안자다. Such "-도 안 VERB-고" is a common phrase describing not doing something that is normally expected (점심도 안 먹고 게임을 했다 = I played game (even) skipping lunch), 공부도 안 하고 시험을 쳤다 = I took the test without studying at all). It puts a (small) emphasis on the action, but it's also a habitual and idiomatic phrase used a lot because it sounds natural. I don't think it is anything that necessarily deserves a special mention. I would simply translate it as: I am staying up late into the night studying English. Feel free to ask again if this doesn't answer your question.