Eftir langan dag er öxlin mín þreytt og mér er illt í hnénu.

Because eftir means after here in a time sense, and in that use it takes the accusative case.

So:

• dagur becomes dag
• langur becomes langan

That gives eftir langan dag = after a long day.

Icelandic is a verb-second language in main clauses. That means the finite verb usually comes in the second position.

Here, Eftir langan dag is placed first as a time phrase, so er comes next:

• Eftir langan dag er öxlin mín þreytt

A more neutral order would be:

• Öxlin mín er þreytt eftir langan dag

Both are possible, but the given sentence follows the normal Icelandic word-order rule.

Öxlin is the definite nominative singular form of öxl, which means shoulder.

Icelandic usually adds the definite article to the end of the noun:

• öxl = a shoulder
• öxlin = the shoulder

The noun öxl is feminine.

In everyday Icelandic, possessives are very often placed after the noun, especially when the noun is definite:

• öxlin mín = my shoulder
• bókin mín = my book

A form like mín öxl is possible, but it sounds much more emphatic, contrastive, or stylistically marked. For ordinary speech, öxlin mín is the natural choice.

Also, mín agrees with öxlin in gender, number, and case, so it is feminine singular nominative here.

It agrees with öxlin mín in gender and number, so it is feminine singular.

The adjective þreyttur has these relevant forms:

• masculine: þreyttur
• feminine: þreytt
• neuter: þreytt

So öxlin mín er þreytt is correct.

Also, after vera, predicate adjectives normally appear in the strong form, even if the noun is definite. That is why you get:

• öxlin er þreytt = the shoulder is tired

but:

• þreytta öxlin = the tired shoulder

Mér is dative.

Icelandic often uses the dative for the person who experiences a physical or emotional state, especially in impersonal expressions like:

• mér er kalt = I am cold
• henni er heitt = she is hot
• honum er illt = he is in pain

So mér er illt literally works like to me it is painful.

In expressions of pain or discomfort, Icelandic often uses a neuter singular adjective as a kind of default form.

So mér er illt is the normal idiomatic pattern for I am in pain / it hurts.

Here, illt is not agreeing with mér. That is why you do not get illur or ill in this sentence.

Not exactly.

Here, illt means something like painful, hurting, or bad in a physical-sensation sense. It is part of the expression mér er illt.

If you want to say I am ill/sick in the English sense, Icelandic more often uses:

• ég er veikur if the speaker is male
• ég er veik if the speaker is female

So mér er illt is about pain, not general illness.

Because í takes the dative when it means location or position, and here the pain is located in the knee.

So the noun hné appears in the dative singular definite form:

• í hné = in a knee
• í hnénu = in the knee

That is why the sentence has í hnénu.

With body parts, Icelandic often uses just the definite noun when the owner is already clear from context.

Since mér already tells you whose knee is meant, í hnénu sounds very natural.

You can say í hnénu mínu, but that is more explicit and often more emphatic, as if you are stressing my knee in particular.

That is mostly a matter of idiomatic choice and what the speaker wants to make explicit.

In the first clause, öxlin mín is the subject, so naming it with a possessive sounds natural. In the second clause, the dative mér already identifies the person affected, so Icelandic often leaves the possessive out with body parts.

So the sentence combines two very normal patterns:

• öxlin mín er þreytt
• mér er illt í hnénu

You could make a sentence like that, and it would probably be understood, but it is not the most natural way to talk about pain in Icelandic.

For aches and pains, Icelandic strongly prefers the pattern:

• mér er illt í ...

So:

• mér er illt í hnénu = normal, idiomatic Icelandic
• hnéð mitt er þreytt = understandable, but less natural for expressing pain

In other words, Icelandic usually describes the experience of pain rather than making the body part itself the subject.