Lekaleitin byrjar núna.

It is a compound noun:

• leki = a leak
• leka = the genitive form of leki
• leit = search, hunt
• -in = the

So lekaleitin is literally something like the leak-search.

Because Icelandic makes compound nouns very freely, and they are normally written as one word.

So where English might use two words, or sometimes a hyphen, Icelandic often prefers a single compound:

• lekaleit = leak search
• lekaleitin = the leak search

The last part of the compound is the main noun, and the earlier part describes it.

It is the definite article, meaning the.

In Icelandic, the is usually attached to the end of the noun instead of standing as a separate word.

So:

• lekaleit = leak search
• lekaleitin = the leak search

Because the head noun is leit, which is a feminine noun.

In Icelandic, the form of the suffixed definite article depends on:

• gender
• number
• case

Here the word is:

• feminine
• singular
• nominative

So leit becomes leitin, and the whole compound becomes lekaleitin.

It is in the nominative singular.

That is because it is the subject of the sentence:

• Lekaleitin = the thing that starts
• byrjar = starts

Subjects in Icelandic are very often in the nominative.

Byrjar is the present tense, 3rd person singular form of byrja, meaning to begin / to start.

It is singular because the subject, lekaleitin, is singular.

So:

• ég byrja = I start
• þú byrjar = you start
• hún byrjar = she starts
• lekaleitin byrjar = the leak search starts

No. Icelandic verbs agree mainly with person and number, not with gender.

So even though lekaleitin is feminine, that does not change the form of byrjar. The important thing here is that the subject is singular, so the verb is 3rd person singular.

That is a very normal word order in Icelandic:

• subject + verb + adverb
• Lekaleitin byrjar núna.

But Icelandic is flexible, especially because main clauses usually follow a verb-second pattern.

You can also say:

• Núna byrjar lekaleitin.

That puts more focus on now.

Both can mean now.

Very roughly:

• nú = now
• núna = now, right now

In many sentences they are interchangeable, and both sound natural.
So you could also say:

• Lekaleitin byrjar nú.

Núna often feels a little more conversational or immediate.

A rough English-friendly approximation is:

LEH-ka-lay-tin BIR-yar NOO-na

A few helpful points:

• Stress usually falls on the first syllable in Icelandic words.
• j in byrjar sounds like y in yes.
• ú sounds like oo.

This is only an approximation, but it is a useful starting point.

Only because it is the first word of the sentence.

Icelandic does not capitalize ordinary nouns the way German does.
So in the middle of a sentence, it would normally be written lekaleitin, not Lekaleitin.