Jeg kan ikke finde nummeret på billetten.

Because Danish usually adds the definite article to the end of the noun instead of putting a separate word in front.

• nummer = number / a number
• nummeret = the number

The ending -et is the definite ending for many neuter nouns.

So in this sentence, nummeret means the number, not just a number.

For the same reason: Danish often puts the at the end of the noun.

• billet = ticket / a ticket
• billetten = the ticket

Here the definite ending is -en, which is common for common-gender nouns.

So:

• nummeret = the number
• billetten = the ticket

Because Danish nouns have grammatical gender.

There are two genders in modern Danish:

• common gender
• neuter

The gender affects:

• the indefinite article
  
  • en billet = a ticket
  • et nummer = a number
• the definite ending
  
  • billetten = the ticket
  • nummeret = the number

So if you learn a noun, it is best to learn it with its article:

• en billet
• et nummer

In Danish main clauses, ikke usually comes after the finite verb.

Here the finite verb is kan.

So the pattern is:

• Jeg kan ikke finde ...

Compare:

• Jeg kommer ikke = I am not coming
• Han vil ikke spise = He does not want to eat

This is very normal Danish word order.

Kan is the present tense of kunne, and here it means can / am able to.

So:

• Jeg kan ikke finde ... = I can't find ...

It can express ability, possibility, or practical success depending on context. In this sentence, it usually means something like:

• I can't find the number
• I’m unable to locate the number

Because it comes after the modal verb kan.

After modal verbs such as:

• kan = can
• vil = want to / will
• skal = must / have to
• må = may / must

the next verb is usually in the infinitive form.

So:

• kan finde = can find
• vil finde = want to find
• skal finde = have to find

That is why it is finde, not finder.

På usually means on, and in this sentence it means the number is on the ticket.

So:

• nummeret på billetten = the number on the ticket

This is a very natural way to say it in Danish when information is printed on a physical object.

For example:

• navnet på listen = the name on the list
• datoen på brevet = the date on the letter

English sometimes uses on, and Danish does too.

Yes. It is one noun phrase meaning the number on the ticket.

The core noun is:

• nummeret = the number

Then på billetten adds extra information and tells you which number:

• the number on the ticket

So the structure is:

• nummeret
  • på billetten

This works much like English noun phrases with prepositional phrases.

Yes, but it means something slightly different.

• Jeg kan ikke finde nummeret på billetten = I can't find the number on the ticket
  • emphasis on inability or failure to locate it
• Jeg finder ikke nummeret på billetten = I am not finding the number on the ticket / I don't find the number on the ticket
  • less natural in many situations if you mean I can't locate it right now

In everyday Danish, if you mean I’m unable to find it, kan ikke finde is the more natural choice.

Because Danish often expresses definiteness by attaching it to the noun.

English:

• the number
• the ticket

Danish:

• nummeret
• billetten

This is one of the biggest differences English speakers notice. Danish does have separate determiners in some situations, but in a simple phrase like this, the definite ending on the noun is the normal pattern.

A rough learner-friendly version is:

• yai kan eg-ge fin-eh noo-mer-ed po bil-eh-dn

A few important notes:

• jeg is often pronounced more like yai than the spelling suggests
• ikke is often pronounced something like eg-ge
• nummeret and billetten often have softer, reduced endings in natural speech
• spoken Danish reduces many unstressed vowels

If you want to sound natural, focus especially on these three words:

• jeg
• ikke
• billetten

They are often quite different from what an English speaker expects from the spelling.

Yes, it could, depending on context. Nummer is a broad word meaning number.

It can refer to:

• a ticket number
• a seat number
• a reference number
• a phone number
• a song/track number in some contexts

In this sentence, på billetten tells us it is the number on the ticket, so it is probably a ticket number, booking number, seat number, or something similar.