Dóttir mín vill hoppa á trampólíninu eftir leikskóla.

In Icelandic, possessives can come after the noun very naturally, especially in everyday neutral language.

• dóttir mín = my daughter
• mín dóttir is possible, but it usually sounds more emphatic, contrastive, or a bit more literary.

So in a plain statement like this, dóttir mín is the most natural choice.

Because the possessive must agree with the noun it belongs to in gender, number, and case.

Here:

• dóttir is feminine
• singular
• nominative (because it is the subject of the sentence)

So the correct form is mín.

Compare:

• sonur minn = my son
• dóttir mín = my daughter
• barn mitt = my child

Kinship words and family terms often behave a little differently in Icelandic.

With words like:

• móðir = mother
• faðir = father
• dóttir = daughter
• sonur = son

it is very common to say:

• dóttir mín
• sonur minn

without adding the definite article ending.

So even though English says my daughter, Icelandic does not need a separate the-type ending here.

Vill is the 3rd person singular present form of the verb vilja = to want.

That matches dóttir mín = my daughter, which is a she/he/it singular subject.

A few forms:

• ég vil = I want
• þú vilt = you want
• hann / hún vill = he / she wants
• við viljum = we want

So dóttir mín vill means my daughter wants.

After vilja = to want, Icelandic normally uses a bare infinitive.

So:

• vill hoppa = wants to jump

not vill að hoppa.

This is similar to English, where want is followed by an infinitive, though English uses to. Icelandic often does not use að after modal-like verbs such as vilja.

Trampólíninu is the definite form of trampólín = trampoline.

So:

• trampólín = a trampoline / trampoline
• trampólíninu = the trampoline

The ending -inu here shows that the noun is:

• singular
• definite
• dative
• neuter

So á trampólíninu means on the trampoline.

This is a very common Icelandic case pattern with prepositions.

With á, the case often depends on whether you mean:

• location = where something happens → dative
• direction / movement onto something → accusative

So:

• hoppa á trampólíninu = jump on the trampoline, meaning the jumping happens there
• hoppa á trampólínið = jump onto the trampoline

In your sentence, the idea is that she wants to be on it and jump there, so dative is used.

After the preposition eftir in a time expression like this, Icelandic uses the dative.

The noun is leikskóli = kindergarten / preschool.

Its singular forms include:

• leikskóli = nominative
• leikskóla = accusative
• leikskóla = dative

So in this sentence, leikskóla is functioning as dative, even though it happens to look the same as the accusative form.

Because this works a lot like English after school.

It refers to the general routine or period of the day, not to a specific the kindergarten as a particular object.

So:

• eftir leikskóla = after kindergarten / after preschool

This is an idiomatic time expression. Icelandic often leaves out the article in that kind of phrase.

Yes, but Icelandic has verb-second word order, so the finite verb usually stays in the second position of the clause.

The neutral order here is:

• Dóttir mín vill hoppa á trampólíninu eftir leikskóla.

You can also front the time phrase:

• Eftir leikskóla vill dóttir mín hoppa á trampólíninu.

Notice that when eftir leikskóla moves to the front, vill still stays in the second slot.

So the word order is flexible, but not completely free.