Hún leitar að lyklunum í handtöskunni sinni.

Because the verb leita usually takes the pattern leita að + dative when it means look for / search for.

So in this sentence:

• leitar = looks for / is looking for
• að lyklunum = for the keys

Here, að is not the English-style to. It is just part of the construction that this verb uses.

A useful thing to memorize is:

• leita að einhverju = look for something
• leita að einhverjum = look for someone

So the case after að here is important: it must be dative.

Because lyklunum is dative plural definite, and that is the form required by leita að.

The noun is lykill = key.

Here are some important plural forms:

• lyklar = keys, nominative plural
• lykla = keys, accusative plural
• lyklum = keys, dative plural
• lyklunum = the keys, dative plural

Since leita að requires the dative, and the meaning is the keys, you get:

• að lyklunum = for the keys

So the ending -unum is doing two jobs at once:

• showing plural
• showing definite (the)
• and the whole form is in the dative

In Icelandic, the is usually not a separate word. It is often attached to the end of the noun as a suffix.

So:

• lyklar = keys
• lyklarnir = the keys
  but in this sentence we need the dative plural:
• lyklum = keys
• lyklunum = the keys

And similarly:

• handtaska = handbag
• handtaskan = the handbag
• handtösku = handbag, accusative/dative singular stem form
• handtöskunni = the handbag, dative singular

So the sentence does contain the, but it is built into the noun endings.

Because the preposition í can take two different cases depending on the meaning:

• dative for location: in
• accusative for motion into: into

Here the meaning is location: she is looking for the keys in her handbag.

So you get:

• í handtöskunni = in the handbag
  dative, because it is a place

Compare:

• Hún er í handtöskunni = she is in the handbag
  location, dative
• Hún setur það í handtöskuna = she puts it into the handbag
  movement into, accusative

That location-vs-motion contrast is very common in Icelandic.

Sinni is the reflexive possessive word meaning her own, his own, or its own, depending on the subject.

In this sentence, it refers back to hún:

• í handtöskunni sinni = in her own handbag

So sinni tells you that the handbag belongs to the subject of the sentence.

This is different from English, where we often just say her and let context do the work.

Because Icelandic usually uses the reflexive possessive sinn when the possessor is the same as the subject of the sentence.

Here:

• subject = hún
• owner of the handbag = the same person

So Icelandic prefers:

• handtöskunni sinni = her own handbag

If you said handtöskunni hennar, that often suggests someone else’s handbag, not the subject’s own.

So the contrast is roughly:

• Hún leitar ... í handtöskunni sinni = she is looking ... in her own handbag
• Hún leitar ... í handtöskunni hennar = she is looking ... in her handbag, meaning some other woman’s handbag

That distinction is very important in Icelandic.

Because sinn has to agree with the noun it belongs to, not with the person who owns the thing.

Here the possessed noun is:

• handtöskunni

That noun is:

• feminine
• singular
• dative

So the reflexive possessive must also be:

• feminine
• singular
• dative

That gives sinni.

This is a very common point of confusion for English speakers. Even though the owner is hún (she), the form of sinn is chosen to match handtöskunni, not hún.

The Icelandic present tense often covers both:

• a general present: she looks for
• an ongoing present: she is looking for

So Hún leitar að lyklunum ... can naturally mean She is looking for the keys ... in context.

If you want to make the ongoing action more explicit, Icelandic can also use:

• Hún er að leita að lyklunum ...

That is closer to She is looking for the keys ... in a more explicitly progressive sense.

But the simple present is completely normal here.

The given sentence has a very normal word order:

• Hún = subject
• leitar = verb
• að lyklunum = object/complement of the verb
• í handtöskunni sinni = prepositional phrase

So:

• Hún leitar að lyklunum í handtöskunni sinni.

But Icelandic word order is fairly flexible, as long as the verb-second pattern is respected in main clauses.

For example, you could also say:

• Í handtöskunni sinni leitar hún að lyklunum.

That puts more emphasis on in her handbag.

So the meaning stays similar, but the focus changes.

A rough guide is:

• Hún ≈ hoon
• leitar ≈ LAY-tar
• að ≈ a soft ath sound, like voiced th in this
• lyklunum ≈ roughly lih-kluh-num or lihk-luh-num
• í ≈ ee
• handtöskunni ≈ HAHNT-tuh-skuhn-nih
• sinni ≈ SIN-nih

A few pronunciation notes:

• ú is like oo in food
• ei is like ay in say
• ð is like th in this
• í is like ee
• double consonants often affect the sound and length of the preceding vowel

The hardest word here for many learners is lyklunum, because consonant clusters in Icelandic can feel unfamiliar. Listening to native audio is especially helpful for that one.