gukjaga eodie issneunji ara?

-가 is the subject marker here. In this sentence, 국자 is the thing whose location is being asked about, so it becomes the subject of the embedded clause:

• 국자가 어디에 있는지 = where the ladle is

Why not -는?

• 국자는 would add a topic nuance, like as for the ladle...
• 국자가 is more neutral and natural when simply asking about its location

Why not -을?

• -을/를 marks a direct object, but 국자 is not the object of 알다 here
• The object of 알다 is really the whole clause 국자가 어디에 있는지 = where the ladle is

So the structure is basically:

• [국자가 어디에 있는지] 알아?
• Do you know [where the ladle is]?

어디 means where, and -에 is a location particle.

So:

• 어디에 = where / in what place / at what location

It is used because 있다 is a verb of existence/location, and Korean normally marks the location with -에:

• 책이 책상 위에 있어요 = The book is on the desk
• 국자가 어디에 있어요? = Where is the ladle?

In casual speech, people sometimes drop -에 and just say 어디, but 어디에 is fully correct and clear.

있는지 can be broken down like this:

• 있다 = to exist / to be located
• 있는 = the present-tense modifier form of 있다
• -지 in -는지 = marks an embedded question, similar to whether or where...

In this sentence, 있는지 is part of the embedded question:

• 국자가 어디에 있는지 = where the ladle is

A useful way to think about -는지 is:

• direct question: 국자가 어디에 있어? = Where is the ladle?
• embedded question: 국자가 어디에 있는지 알아? = Do you know where the ladle is?

So -는지 is what turns the question into something that can be the object of another verb like 알다.

Because this is not a direct question. It is an embedded question.

Compare these:

• 국자가 어디에 있어? = Where is the ladle?
  This is a direct question.

• 국자가 어디에 있는지 알아? = Do you know where the ladle is?
  This is an indirect or embedded question.

Korean commonly uses -는지 for embedded questions after verbs like:

• 알다 = to know
• 모르다 = to not know
• 궁금하다 = to wonder / be curious
• 묻다 = to ask

Examples:

• 그가 언제 오는지 알아요? = Do you know when he is coming?
• 뭐가 문제인지 모르겠어요 = I don't know what the problem is

So 있는지 is exactly what you want when you mean where it is inside a larger sentence.

알아? is the casual, non-polite present-tense form of 알다 = to know.

So:

• 알아? = Do you know?

This tells you the sentence is in an informal speaking style, probably used with:

• a friend
• a younger person
• someone close to the speaker

More polite versions would be:

• 국자가 어디에 있는지 알아요?
• 국자가 어디에 있는지 아세요?

The difference between those two:

• 알아요? = polite, neutral
• 아세요? = polite and honorific, often used when showing respect to the listener

Korean very often leaves out subjects and objects when they are understood from context.

So 알아? literally just means know?, but in natural English we translate it as:

• Do you know?

The subject you is omitted because Korean does not need to say it if it is obvious from the situation.

This is extremely common in Korean. For example:

• 가? = Are you going?
• 먹었어? = Did you eat?
• 알아? = Do you know?

So the sentence is perfectly complete even without 너는.

The structure is:

• [embedded question] + 알아?

More specifically:

• 국자가 = the ladle + subject marker
• 어디에 = where
• 있는지 = is / is located + embedded question marker
• 알아? = do you know?

So the sentence is:

• [국자가 어디에 있는지] 알아?
• Do you know [where the ladle is]?

A very important Korean pattern is:

• who/what/where/when + ... + -는지 알다

Examples:

• 그 사람이 누구인지 알아? = Do you know who that person is?
• 회의가 언제인지 알아요? = Do you know when the meeting is?

Yes, in casual conversation, Koreans often drop particles if the meaning is still clear.

So these can all be heard:

• 국자가 어디에 있는지 알아?
• 국자 어디에 있는지 알아?
• 국자 어디 있는지 알아?

The full version with particles is the clearest for learners and is grammatically complete. The shorter versions sound more conversational.

Still, there are small nuances:

• 국자가 어디에 있는지 알아? = full and clear
• 국자 어디 있는지 알아? = casual, natural in speech

As a learner, it is a good idea to first get comfortable with the full version before relying too much on particle omission.

Here it means to be located / to be in a place.

있다 has a few related uses, including:

• to exist
• to be present
• to be located
• to have in some contexts

In this sentence, because it is used with 어디에 and asks about location, the meaning is clearly:

• to be located

So:

• 국자가 어디에 있다 = the ladle is somewhere / the ladle is located somewhere

That is why the natural English translation uses is rather than something like exists.

You can change the ending depending on politeness level.

Common versions:

• 국자가 어디에 있는지 알아?
  casual

• 국자가 어디에 있는지 알아요?
  polite casual

• 국자가 어디에 있는지 아세요?
  polite with honorific nuance toward the listener

• 국자가 어디에 있는지 압니까?
  formal, more stiff or official

In daily conversation, 알아요? or 아세요? is much more common than 압니까?

So if you are unsure what to use, 국자가 어디에 있는지 알아요? is a safe and natural choice.