Hún lofaði að redda nýju lyklakorti fyrir mig á morgun.

Lofaði is the past tense of lofa = to promise.

So:

• hún lofar = she promises
• hún lofaði = she promised

In this sentence, hún tells you the subject is she, and lofaði shows the action happened in the past.

Here að is the infinitive marker, like English to.

So:

• lofa að gera eitthvað = to promise to do something
• hún lofaði að redda... = she promised to sort out/get...

This is a very common pattern in Icelandic:
verb + að + infinitive

Redda is a very common everyday Icelandic verb, often a bit informal/colloquial. Its meaning depends on context, but it often means things like:

• fix
• sort out
• arrange
• get hold of
• manage to provide

In this sentence, it means something like sort out / get hold of / arrange a new keycard.

So it is less like a formal provide, and more like everyday I’ll sort it out.

Because redda takes a dative object.

The noun lyklakort is neuter singular, and in the dative it becomes:

• lyklakort → lyklakorti

The adjective nýr has to agree with the noun in gender, number, and case, so in dative singular neuter it becomes:

• nýtt → nýju

So:

• nýtt lyklakort = nominative/accusative form
• nýju lyklakorti = dative form

That is why the sentence uses nýju lyklakorti.

Because Icelandic normally has no indefinite article like English a/an.

So nýju lyklakorti can mean a new keycard without any extra word.

Compare:

• lyklakort = a keycard / keycard
• lyklakortið = the keycard

The definite article is usually added to the end of the noun, but the indefinite idea is usually shown by having no article at all.

Because here fyrir means for me / on my behalf, and with that meaning it takes the accusative, so:

• mig = accusative of ég
• mér = dative of ég

So:

• fyrir mig = for me

English speakers often want to use mér because me feels indirect, but after prepositions you have to learn the case that the preposition requires in that meaning.

In this sentence, á morgun most naturally goes with redda.

So the idea is:

• she promised
• that she would sort out the new keycard tomorrow

That reading is natural because á morgun comes at the end of the infinitive phrase að redda nýju lyklakorti fyrir mig á morgun.

Because Icelandic uses compound nouns very freely.

So lyklakort is a compound:

• lykill = key
• kort = card
• lyklakort = keycard

This is very normal in Icelandic. Where English might write two words, a hyphen, or one word depending on the expression, Icelandic often prefers one compound word. The first part may also change shape slightly inside the compound, which is why you get lykla- rather than the full dictionary form lykill.