Undskyld, har De en kuglepen, jeg må låne, hvis min egen ikke virker?

De means you in a formal, polite sense.

A few useful points:

• De = formal you
• du = informal you
• It is traditionally capitalized to show respect and to distinguish it from de, which usually means they

So Har De ...? is like saying Do you have ...? in a very polite way.

In modern everyday Danish, most people would normally say du.

De is:

• very formal
• somewhat old-fashioned
• sometimes used in very polite service language, official situations, or when speaking to older people in a very respectful way

So this sentence is understandable and correct, but it sounds more formal than everyday speech. A more typical modern version would be:

Undskyld, har du en kuglepen, jeg må låne, hvis min egen ikke virker?

Because Danish yes/no questions usually use verb-first word order.

Compare:

• Statement: De har en kuglepen. = You have a pen.
• Question: Har De en kuglepen? = Do you have a pen?

So the verb har moves before the subject De to make it a question.

Undskyld can mean sorry, excuse me, or pardon me, depending on context.

Here it means something like:

• Excuse me
• Sorry to bother you

It is a natural way to politely get someone's attention before asking for something.

More specifically, kuglepen means ballpoint pen.

In everyday Danish, people also often just say:

• en pen = a pen

So both are possible, but kuglepen is more specific. If you want to say pencil, that is blyant.

Because after a modal verb in Danish, the next verb stays in the infinitive.

Here:

• må = may / am allowed to
• låne = borrow

So:

• jeg må låne = I may borrow
• not jeg må låner

This is the same pattern you get with other modal verbs:

• jeg kan komme = I can come
• jeg vil gå = I want to go
• jeg skal arbejde = I have to work

må is used because the speaker is asking for permission.

So:

• må jeg låne ...? = may I borrow ...?
• kan jeg låne ...? can also be heard, but it leans more toward can/am I able to

In English, can I borrow... is very common, but in Danish, må jeg låne... is a very natural polite way to ask.

Because Danish can leave out the relative word som in cases like this.

The fuller version would be:

en kuglepen, som jeg må låne

That means:

• a pen that I may borrow

In everyday Danish, leaving out som is common when it is not the subject of the clause. So both versions are understandable, but the version without som is quite natural.

egen means own, so min egen means my own.

This adds emphasis and contrast:

• min = my
• min egen = my own

So the sentence is specifically talking about the speaker's own pen not working.

You could say just min, and the meaning would still be clear in context, but min egen sounds a bit more explicit.

Because hvis introduces a subordinate clause, and subordinate clauses in Danish have different word order from main clauses.

Compare:

• Main clause: Min egen virker ikke.
• Subordinate clause: hvis min egen ikke virker

So in a subordinate clause, the negation ikke comes before the finite verb:

• ... hvis min egen ikke virker

This is a very important Danish pattern.

Yes. virker means works / functions.

It is completely natural to use virke about a pen, machine, app, button, and so on.

So:

• Min pen virker ikke = My pen doesn't work

You may also hear skriver ikke for a pen:

• Min pen skriver ikke = My pen doesn't write

That is a bit more specific, but virker ikke is perfectly natural.

There are commas because Danish punctuation often marks inserted and subordinate parts of the sentence clearly.

Here:

• Undskyld, — comma after the introductory word
• en kuglepen, jeg må låne, — commas around the inserted relative clause
• hvis min egen ikke virker — a subordinate clause

One extra note: Danish has had more than one accepted comma system, so you may sometimes see fewer commas in modern texts. But the commas in your sentence are completely normal and easy for learners to encounter.

Yes, it is correct and understandable, but it sounds rather formal because of De.

A more everyday version would be:

Undskyld, har du en pen, jeg må låne, hvis min ikke virker?

And if your pen already does not work, many Danes might say something even simpler, such as:

Undskyld, må jeg låne en pen? Min virker ikke.

So the original sentence is fine, but slightly more careful and formal than typical casual speech.