Henni tókst að finna reikningsnúmerið mitt eftir að hún las gamla tölvupóstinn.

Because the verb takast uses a special construction:

eitthverjum tekst / tókst að + infinitive

That means someone manages / managed to do something, but the someone is put in the dative case, not the nominative.

So:

• hún = nominative, used for an ordinary subject
• henni = dative, required by tókst

So Henni tókst að finna... means literally something like To her, it succeeded to find..., which Icelandic uses for She managed to find...

Tókst is the past tense of takast.

A very common pattern is:

• Mér tekst að... = I manage to...
• Mér tókst að... = I managed to...

So here:

• Henni tókst = she managed / she succeeded

Even though it translates naturally with she, grammatically the person is still in the dative case because of how takast works.

After takast, Icelandic normally uses að + infinitive to say what someone managed to do.

So:

• tókst að finna = managed to find
• tókst að lesa = managed to read
• tókst að koma = managed to come

Here, finna is the infinitive to find.

It has three parts:

• reikningsnúmer = account number
• -ið = the definite article attached to the noun
• mitt = my

So:

• reikningsnúmer = an account number
• reikningsnúmerið = the account number
• reikningsnúmerið mitt = my account number

This is very normal in Icelandic: with a possessive pronoun after the noun, the noun is usually definite.

A literal breakdown would be something like:

• account-number-the my

But the natural English meaning is simply my account number.

Because mitt has to agree with reikningsnúmerið.

The noun reikningsnúmer is:

• neuter
• singular

So the possessive pronoun must also be:

• neuter
• singular

That gives:

• masculine: minn
• feminine: mín
• neuter: mitt

Since reikningsnúmer is neuter, mitt is the correct form.

That is the normal Icelandic pattern.

Very often Icelandic says:

• bíllinn minn = my car
• húsið mitt = my house
• reikningsnúmerið mitt = my account number

So the usual structure is:

noun + definite article + possessive pronoun

Putting the possessive before the noun is possible in some contexts, but it is less neutral and often gives emphasis or a slightly different feel. For a basic sentence like this, reikningsnúmerið mitt is the standard way.

Eftir að means after when it introduces a full clause.

So:

• eftir að hún las gamla tölvupóstinn = after she read the old email

This is different from eftir used with just a noun phrase, for example:

• eftir mat = after dinner

Because this sentence has a whole clause with a subject and verb — hún las — Icelandic uses eftir að.

Las is the past tense of lesa, meaning to read.

So:

• lesa = to read
• les = read / am reading
• las = read, in the past

This verb is irregular, so you just need to learn that:

• hún las = she read

Because this phrase is the direct object of las, so it must be in the accusative case.

The base forms are:

• gamall = old
• tölvupóstur = email

But here the noun is:

• masculine
• singular
• definite
• accusative

That changes the forms to:

• gamla = weak adjective form
• tölvupóstinn = the email, accusative definite

So:

• gamall tölvupóstur = an old email, nominative
• gamla tölvupóstinn = the old email, accusative

The adjective is weak because the noun is definite.

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

Here the first element is Henni, so the verb comes next:

• Henni tókst að finna...

That is perfectly normal Icelandic word order.

The sentence could be analyzed like this:

• Henni = dative experiencer
• tókst = finite verb
• að finna reikningsnúmerið mitt = infinitive phrase
• eftir að hún las gamla tölvupóstinn = time clause

So the order follows normal Icelandic syntax.

Yes. They both refer to the same woman, but they appear in different cases because they do different grammatical jobs.

• henni = dative, required by tókst
• hún = nominative, the ordinary subject of las

So in English both refer to she, but Icelandic changes the form depending on case.

Yes, that is a very natural way to understand it.

The core idea of henni tókst að finna is:

• she managed to find
• she succeeded in finding
• she was able to find

And the final clause explains when or how that became possible:

• after she read the old email

So the sentence suggests that reading that email gave her the information she needed to find the account number.