I'm confused about what sentence to use
"Late time" is a combination that you find in science nowadays. It is sometimes found to express what you want to say in your sentence but not as often as "late hour" (ngram).
- Why are you still awake at this late hour?
As mentioned by user John Lawler in the comments, you can also do away with the adjective "late" and say "Why are you still awake at this hour?"; that is saying the same thing.
You can say instead "this late time of (the) day/the morning/the year" (ngram) (also "this early time of (the) day/the morning/the year" for the opposite).