Hún speglar sig í speglinum og lagar hárið sitt.

Because að spegla sig is a reflexive expression in Icelandic. It means to look at oneself in a mirror / to mirror oneself.

• speglar = looks at / mirrors
• sig = oneself / herself / himself / themselves depending on the subject

So Hún speglar sig means that she is looking at herself, not at someone else.

Sig is the reflexive pronoun used in the third person.

It refers back to the subject of the clause:

• Hún speglar sig = She looks at herself

You use sig when the person doing the action and the person receiving the action are the same.

Compare:

• Hún speglar sig = She looks at herself
• Hún speglar hana = She looks at her (some other female person)

So sig is important because it shows that the action comes back to the subject.

Because í can take either dative or accusative, depending on meaning.

• dative = location, in / inside / at
• accusative = motion into something

Here the meaning is location: she is looking at herself in the mirror, not moving into it. So Icelandic uses the dative:

• í speglinum = in the mirror

Compare:

• Hún er í speglinum = She is in the mirror / reflected in the mirror
• Hún setur mynd í spegilinn would imply motion toward/into something, so accusative would be used in a different kind of sentence

So the short answer is: location → dative → speglinum.

That ending shows two things at once:

1. the noun is definite: the mirror
2. it is in the dative singular

The base noun is:

• spegill = mirror

In this sentence, after í with a location meaning, it becomes dative singular definite:

• speglinum = the mirror (dative singular)

Icelandic usually adds the definite article onto the end of the noun rather than using a separate word like English the.

Because the hair belongs to the subject of the sentence, and Icelandic normally uses the reflexive possessive in that situation.

• sinn / sína / sitt = one’s own / his own / her own / their own
• hennar = her (not reflexive)

So:

• Hún lagar hárið sitt = She fixes her own hair
• Hún lagar hárið hennar = She fixes her hair, meaning another woman’s hair

That is a very important distinction in Icelandic. Sitt points back to the subject; hennar does not.

Because sinn/sína/sitt agrees with the noun being possessed, not with the owner.

Here the possessed noun is:

• hárið = the hair

Hár is a neuter noun, and here it is singular, so the correct possessive form is:

• sitt = neuter singular

Compare:

• bílinn sinn = his/her own car (bíll is masculine)
• töskuna sína = his/her own bag (taska is feminine)
• hárið sitt = his/her own hair (hár is neuter)

So even though the subject is hún (she), the form is sitt because it matches hárið.

In Icelandic, it is completely normal to have:

• a definite noun
• plus a possessive

So hárið sitt literally looks like the hair her-own, but that is normal Icelandic grammar.

English usually says just her hair, without the, but Icelandic often uses the definite form in expressions like this.

So:

• lagar hárið sitt = fixes her hair / her own hair

This is not redundant in Icelandic; it is the natural way to say it.

It is the direct object of lagar, so it is in the accusative.

The verb:

• að laga = to fix / adjust / arrange

takes a direct object, and here that object is hárið.

A useful detail: for many neuter nouns, the nominative and accusative singular forms are the same, so hárið looks the same in both cases. But functionally in this sentence, it is accusative because it is the object of the verb.

They are both present tense forms:

• hún speglar = she looks at herself / is looking at herself
• hún lagar = she fixes / is fixing

In Icelandic, the present tense can often cover both:

• a general/habitual meaning: she looks at herself and fixes her hair
• an action happening now: she is looking at herself and fixing her hair

The exact English translation depends on context.

Because Icelandic, like English, often leaves out the repeated subject in a coordinated sentence when it is clearly the same subject.

So:

• Hún speglar sig í speglinum og lagar hárið sitt.

means:

• She looks at herself in the mirror and fixes her hair.

You could think of it as:

• Hún speglar sig í speglinum og [hún] lagar hárið sitt

The second hún is simply omitted because it is understood.

Here are the main dictionary forms:

• hún = she
• speglar → að spegla sig = to look at oneself in a mirror
• sig = reflexive pronoun (oneself / herself / himself)
• í = in
• speglinum → spegill = mirror
• og = and
• lagar → að laga = to fix, arrange, adjust
• hárið → hár = hair
• sitt → sinn / sín / sitt = one’s own

Looking up the dictionary forms can make the sentence much easier to analyze.

Here hárið is singular.

In Icelandic, hár can refer to hair as a mass noun, much like English hair in she fixed her hair. So:

• hárið sitt = her hair

It does not mean a single hair strand here. It means her hairstyle or her hair in general.

So even though English and Icelandic do not always organize mass nouns in exactly the same way, this use is very natural and straightforward in Icelandic.