Eftir því hvort spegillinn er viðkvæmur pökkum við honum inn.

It is an idiomatic expression meaning depending on whether.

• eftir því can mean according to that or depending on that
• hvort means whether

So Eftir því hvort spegillinn er viðkvæmur means depending on whether the mirror is fragile.

This is because Icelandic follows the verb-second rule in main clauses.

In this sentence, the long adverbial phrase Eftir því hvort spegillinn er viðkvæmur comes first. When something other than the subject comes first, the finite verb must come next, and the subject comes after that:

• Eftir því hvort spegillinn er viðkvæmur = first element
• pökkum = finite verb
• við = subject

A more neutral word order would be:

Við pökkum honum inn eftir því hvort spegillinn er viðkvæmur.

There are two reasons:

1. spegill is a masculine noun, so when you refer back to it with a pronoun, Icelandic uses a masculine pronoun.
2. The verb að pakka takes a dative object, not an accusative one.

So:

• honum = dative masculine singular
• hann would be accusative/nominative masculine, so it does not fit here
• það would be neuter, but spegill is not neuter

This is very common in Icelandic: pronouns agree with the grammatical gender of the noun, even when the thing is inanimate.

Because it agrees with spegillinn.

Here, spegillinn is:

• masculine
• singular
• nominative

So the predicate adjective also appears in the masculine singular nominative form:

• spegillinn er viðkvæmur = the mirror is fragile

Compare:

• glasið er viðkvæmt = the glass is fragile
  because glas is neuter

spegillinn is the noun spegill with the suffixed definite article:

• spegill = mirror
• spegillinn = the mirror

It is in the nominative singular, because it is the subject of er in the clause spegillinn er viðkvæmur.

Because hvort means whether, while ef usually means if.

After eftir því, Icelandic normally uses hvort in this kind of structure:

• eftir því hvort ... = depending on whether ...

So this is the standard and natural choice here.

að pakka inn einhverju means to wrap something up or to package something.

The word inn is a particle, similar to particles in English verbs like:

• wrap up
• pack up

So in the sentence:

• pökkum við honum inn

the basic idea is we wrap/package it.

Yes. A straightforward version would be:

Við pökkum honum inn eftir því hvort spegillinn er viðkvæmur.

This means the same thing. The original sentence simply puts the depending on whether... part first, which affects emphasis and triggers the normal verb-second word order.

Yes. You could say:

Við pökkum speglinum inn.

Here speglinum is the dative singular form of spegill, which is required because að pakka takes the dative.

So:

• honum = to/for it, in the dative
• speglinum = the mirror, in the dative

Yes.

The subordinate clause is:

hvort spegillinn er viðkvæmur = whether the mirror is fragile

That clause is introduced by hvort. Together with Eftir því, it forms the larger adverbial phrase:

Eftir því hvort spegillinn er viðkvæmur = depending on whether the mirror is fragile