Hon kan inte bestämma sig för vilken bok hon vill låna.

Because bestämma sig is the usual Swedish expression for to decide / to make up one’s mind.

• bestämma by itself often means decide, determine, or set
• bestämma sig means decide oneself, idiomatically make up one’s mind

So in this sentence:

• Hon kan inte bestämma sig = She can’t make up her mind

The word sig is a reflexive pronoun, like oneself / herself.
Since the subject is hon, the reflexive form is still sig.

Compare:

• Jag bestämmer mig = I decide / make up my mind
• Du bestämmer dig = You decide
• Hon bestämmer sig = She decides

So this is a fixed and very common pattern in Swedish.

Because bestämma sig för is the full expression meaning decide on or choose something.

So:

• bestämma sig = make up one’s mind
• bestämma sig för något = decide on something

In your sentence:

• bestämma sig för vilken bok = decide on which book

This works much like English decide on in some contexts.

Examples:

• Jag måste bestämma mig för en färg. = I have to decide on a color.
• Han har bestämt sig för att resa. = He has decided to travel.

So the för belongs naturally with the verb phrase.

Because vilken means which, and it is used when choosing among known or possible options.

• vilken bok = which book
• vad usually means what

Since the sentence is about choosing one book from some set of books, Swedish uses vilken, not vad.

Also, vilken changes form depending on the noun:

• vilken for common-gender singular nouns
• vilket for neuter singular nouns
• vilka for plural nouns

Here, bok is a common-gender singular noun, so:

• vilken bok = correct

Examples:

• vilken film = which film
• vilket hus = which house
• vilka böcker = which books

Because hon vill låna is a full subordinate clause, and Swedish normally needs the subject to be stated.

The structure is:

• Hon kan inte bestämma sig för
  • vilken bok hon vill låna

That last part literally contains:

• which book she wants to borrow

Just like in English, you need she there.

You cannot normally leave it out and say:

• vilken bok vill låna ❌

That would be missing the subject.

So the second hon is not unnecessary repetition; it is simply the subject of the clause hon vill låna.

Because vill is a modal verb, and modal verbs in Swedish are followed by the infinitive without att.

So:

• hon vill låna = she wants to borrow

This is similar to English:

• she wants to borrow
  not she wants to to borrow

Common Swedish modal verbs include:

• kan = can
• vill = want to
• ska = will / shall / going to
• måste = must
• får = may / be allowed to

Examples:

• Jag kan simma. = I can swim.
• Hon vill läsa. = She wants to read.
• Vi måste gå. = We must go.

With many other verbs, Swedish does use att before the infinitive, but not after modal verbs.

Because Swedish main clauses usually follow V2 word order, meaning the finite verb comes in the second position.

In this sentence, the main clause begins with:

• Hon = subject
• kan = finite verb
• inte = negation

So the order is:

• Hon kan inte ...

That is the normal Swedish order in a main clause.

Compare:

• Hon kan inte bestämma sig. = She cannot decide.
• Jag vill inte gå. = I do not want to go.

If this were a subordinate clause, the word order could be different, with inte before the finite verb:

• ... att hon inte kan bestämma sig = ... that she cannot make up her mind

So this sentence is a good example of normal main-clause order.

This is a very common source of confusion.

By itself, låna usually means borrow:

• Hon vill låna en bok. = She wants to borrow a book.

To say lend, Swedish often uses:

• låna ut = lend out / lend

Examples:

• Kan jag låna din penna? = Can I borrow your pen?
• Kan du låna ut din penna? = Can you lend me your pen?

So in your sentence, låna clearly means borrow.

They are related, but not the same.

bestämma without sig can mean things like:

• decide
• determine
• set
• control

Examples:

• Chefen bestämmer. = The boss decides / is in charge.
• Vi måste bestämma tiden. = We must decide the time.

bestämma sig is more specifically to make up one’s mind.

Example:

• Jag kan inte bestämma mig. = I can’t make up my mind.

So in your sentence, bestämma sig is used because the idea is that she is struggling to choose.

It breaks down like this:

• Hon = she
• kan inte = cannot
• bestämma sig för = decide on / make up her mind about
• vilken bok = which book
• hon vill låna = she wants to borrow

So the structure is:

• She cannot decide on which book she wants to borrow

You can think of it in two parts:

1. Main clause: Hon kan inte bestämma sig för ...
2. Embedded question / subordinate clause: vilken bok hon vill låna

That second part is not a direct question, so Swedish does not use question word order there.

Compare:

• Vilken bok vill hon låna? = Which book does she want to borrow?
  direct question

• Jag vet vilken bok hon vill låna. = I know which book she wants to borrow.
  embedded clause

Your sentence uses the second type.

Yes. A natural alternative is:

• Hon kan inte bestämma vilken bok hon vill låna.

This is also correct and natural.

The version with bestämma sig för feels a bit more like:

• She can’t make up her mind about which book to choose

The version without sig för is often a little more direct:

• She can’t decide which book she wants to borrow

Both are good Swedish, but bestämma sig för is especially useful because it highlights the idea of choosing among options.