jigeum myeot siinji mollayo.

-인지 here turns 몇 시 (“what time”) into an indirect question clause, serving as the object of 모르다 (“to not know”).

• In English this is like “I don’t know what time (it is).”
• Korean verbs of knowing/feeling (알다·모르다·궁금하다·확신하다 등) require a noun clause before them.
• Without -인지, 몇 시 is just a bare question word, not a clause.

Because 모르다 needs a full clause (a noun phrase) as its object, not a standalone question word.

• 몇 시 alone is incomplete.
• 몇 시인지 is that complete clause: “what time it is.”
• Omitting -인지 makes the sentence ungrammatical.

They serve different roles:

• 시 is used for clock time (“o’clock,” e.g. 3시 = “3 o’clock”).
• 시간 means “a length/duration of time” (e.g. 한 시간 = “one hour” duration).
  Since you’re asking for the current clock time, you use 몇 시.

모르다 is a 르-irregular verb. In the polite present tense:

1. Drop 르 from the stem (모르 → 모).
2. Double the preceding consonant ㄹ.
3. Add -아요 → 몰라요.
   So 모르다 + -아요 → 몰라요.

Yes, you can say 지금 몇 시인지 모르겠어요.

• 몰라요 is simply “I don’t know.”
• 모르겠어요 adds a shade of “I’m not sure / I don’t think I know.”
  The second is slightly more tentative or polite.

You can adjust formality by changing the verb ending:

• Formal/Standard: 지금 몇 시인지 모릅니다 or 지금 몇 시인지 잘 모르겠습니다
• Casual/Infimiliar: 지금 몇 시인지 몰라
• Direct question version (if you want to ask instead of state): 지금 몇 시예요?

지금 means “now,” indicating the current moment.

• Position: It’s an adverb and can go at the beginning or before the verb:
  
  • 지금 몇 시인지 몰라요
  • 몇 시인지 지금 몰라요
• Omission: If the context is clear, you can simply say 몇 시인지 몰라요.