Hún finnur fjarstýringuna loksins undir sófanum og kveikir á sjónvarpinu.

Because fjarstýringuna means the remote control as a direct object.

Breakdown:

• fjarstýring = remote control
• It is a feminine noun
• In this sentence, it is the thing she finds, so it is in the accusative
• The ending -una includes the definite article (the)

So:

• fjarstýring = a remote control
• fjarstýringuna = the remote control

Icelandic usually puts the at the end of the noun rather than using a separate word like English does.

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

Here is the rule:

• accusative after undir = movement toward or into position
• dative after undir = location, being under something

In this sentence, the remote is already located there:

• undir sófanum = under the sofa

If you were describing movement, you would use the accusative:

• Hún setur fjarstýringuna undir sófann. = She puts the remote under the sofa.

So:

• undir sófanum = under the sofa, location
• undir sófann = to under the sofa / under the sofa as a destination

Sófanum is the dative singular definite form of sófi (sofa).

Breakdown:

• sófi = sofa
• sófanum = the sofa

Why dative? Because, as explained above, undir takes the dative when it means location.

So the full phrase:

• undir sófanum = under the sofa

This is a very common pattern in Icelandic prepositional phrases.

Loksins means finally or at last.

In this sentence:

• Hún finnur fjarstýringuna loksins...
• literally: She finds the remote control finally...
• natural English: She finally finds the remote control...

Its placement is flexible, but Icelandic word order does not always match English word order exactly. You could often also hear:

• Hún finnur loksins fjarstýringuna...

Both are understandable, but the sentence you were given is perfectly natural.

Because the Icelandic verb kveikja á means to turn on something.

It is a fixed verb pattern:

• kveikja á einhverju = to switch something on

So you must use á here.

Examples:

• kveikja á ljósinu = turn on the light
• kveikja á tölvunni = turn on the computer
• kveikja á sjónvarpinu = turn on the television

Without á, it would not mean the same thing.

Because kveikja á requires the dative case.

Breakdown:

• sjónvarp = television
• It is a neuter noun
• sjónvarpinu = the television in the dative singular definite

So:

• kveikir á sjónvarpinu = turns on the television

This is something learners often need to memorize with the verb:

• kveikja á + dative
• slökkva á + dative = turn off

Historically and literally, yes, á often means on, but in this sentence you should think of kveikja á as a single verb expression meaning turn on.

So it is better to learn:

• kveikja á = turn on
• slökkva á = turn off

Rather than translating each word separately every time.

This is similar to English expressions like turn on, where the meaning depends on the whole phrase, not just the verb alone.

Both verbs are in the present tense:

• finnur = finds
• kveikir = turns on

Icelandic often uses the present tense in examples, summaries, and narration, just like English can:

• She finds the remote and turns on the TV.

If you wanted the past tense, it would be:

• Hún fann fjarstýringuna loksins undir sófanum og kveikti á sjónvarpinu.
• She finally found the remote under the sofa and turned on the television.

The dictionary forms are:

• finna = to find
• kveikja á = to turn on

The forms in the sentence are:

• finnur = present tense of finna
• kveikir = present tense of kveikja

It is useful to learn Icelandic verbs with the pattern they require:

• finna eitthvað = find something
• kveikja á einhverju = turn something on

That helps you remember the correct case and preposition.

Because Icelandic usually uses a suffixed definite article. That means the is attached to the end of the noun.

In this sentence:

• fjarstýringuna = the remote control
• sófanum = the sofa
• sjónvarpinu = the television

This is one of the first big differences English speakers notice. Instead of saying:

• the remote
• the sofa
• the television

Icelandic often says the noun plus a special ending.

It can mean either, depending on context.

In this sentence:

• kveikir á sjónvarpinu most naturally means:
• turns on the TV / television set

Because you physically switch it on.

But sjónvarp can also refer more generally to television as a medium:

• Ég horfi mikið á sjónvarp. = I watch a lot of television.

So context tells you whether it means:

• the device
• television in general

The sentence is:

• Hún = she
• finnur = finds
• fjarstýringuna = the remote control
• loksins = finally
• undir sófanum = under the sofa
• og = and
• kveikir á sjónvarpinu = turns on the television

So the structure is roughly:

Subject + verb + object + adverb + prepositional phrase + and + verb phrase

Very literally:

• She finds the remote control finally under the sofa and turns on the television.

Natural English:

• She finally finds the remote control under the sofa and turns on the TV.