Hun er den eneste, der ved det.

No. Modern Danish does not use separate masculine and feminine articles here. Den is used with common gender nouns, and det with neuter nouns.

In Hun er den eneste, the phrase means she is the only one / person. The implied noun is something like person, which is common gender, so den is correct.

So:

• den eneste = the only one / the only person
• det eneste = the only thing

The natural gender of the person does not change den in this structure.

Because Danish normally needs den before eneste in this kind of expression.

Den eneste works like the only one in English. Even though the noun is not spoken, it is still understood.

Natural Danish:

• Hun er den eneste, der ved det.
• Hun er den eneste person, der ved det.

Without den, the sentence sounds incomplete or nonstandard in ordinary Danish.

Here der introduces a relative clause. In English, it corresponds to who or that.

So:

• den eneste = the only one
• der ved det = who knows it / that knows it

It links the second part of the sentence back to den eneste.

In this sentence, der is the normal choice because it is the subject of the relative clause:

• der ved det = who knows it

A very useful beginner rule is:

• der when the relative word is the subject
• som when it is the object

Compare:

• Manden, der kommer nu = the man who is coming now
• Manden, som jeg kender = the man whom I know

You may hear some variation in real Danish, but den eneste, der ved det is the form learners should expect and use.

Because Danish divides English to know into different verbs.

• at vide = to know a fact, answer, or piece of information
• at kende = to know a person, place, or be familiar with something

Here the sentence is about knowing some information, so Danish uses at vide:

• Hun ved det = she knows it

Compare:

• Jeg kender hende = I know her
• Jeg ved det = I know it

Ved is the present tense of at vide.

Important forms:

• infinitive: at vide
• present: ved
• past: vidste
• past participle: vidst

Examples:

• Hun ved det = she knows it
• Hun vidste det = she knew it

Det means it or that, depending on the context. It refers to some fact or piece of information already known from the conversation.

So ved det means:

• knows it
• knows that

Danish often uses det this way when the listener already knows what is being talked about.

Because der ved det is a relative clause, and Danish normally separates that clause from the main clause with a comma.

So the structure is:

• main clause: Hun er den eneste
• relative clause: der ved det

That is why it is written:

• Hun er den eneste, der ved det.

No, they are related in meaning but used differently.

• eneste = only, in the sense of the only one
• kun = only, in the sense of just / merely

So:

• Hun er den eneste, der ved det. = She is the only one who knows it.
• Hun ved kun det. = She only knows that / that is all she knows.

In your sentence, eneste is correct because the idea is that nobody else knows it.