"Are you still awake?"
1. 아직 깨(어) 있어? (casual) / 아직 깨어 있으세요? (formal) - Are you still awake?
2. 아직 잠 안 들었어? / 아직 잠 안 드셨어요? - Are you still not asleep? (speaking to someone lying in bed)
(자다 means "go to bed" and sometimes "fall asleep". 잠들다 or 잠이 들다 is strictly "fall asleep" or "go to sleep")
깨다 is "wake up", so we can say "잠을 깨다" or "잠에서 깨어나다" (나다 adds the sense of "out").
일어나다 is "get up" or "get out of bed".
Being awake is expressed as 깨(어) 있다, meaning "be in the awake state", the opposite of 잠들었다/잠들어 있다(asleep). 깨어 is its -아/어 connective form, but 어 is often omitted.
Related words are 깨다, 잠들다, 깼다, 잠들었다, 깨 있다, 잠들어 있다, 깨우다, 재우다, 깨어나다, etc.
- 아이가 피곤해서 잠을 깨지를 못한다: The kid can't wake up as he's so tired.
- 내일 아침에 나 깨우지 마: Don't wake me up tomorrow morning (깨우다: wake (someone) - causative of 깨다).
- 10시간이 지난 후에야 마취에서 깨어났다: I awoke from anesthesia after 10 full hours.
- 잠을 깬 채로 누워 있었다: I lay in bed awakened from sleep.
There is also a song titled 잠들지 않는 그리움 (A Sleepless Yearning), with this line:
- 잠이 깨어 있지만 눈을 감고 있었어....I awoke but stayed in bed with eyes closed