La clienta se negó a pagar hasta que le trajeran la cuenta correcta, así que al final la jefa tuvo que disculparse.

Because both refer to women, so Spanish uses feminine forms.

• la clienta = a female customer
• la jefa = a female boss/manager

In Spain, clienta and jefa are very normal. You may also hear la cliente in some places, but clienta is perfectly correct and common.

Negarse a + infinitive is a fixed pattern meaning to refuse to do something.

So:

• se negó a pagar = she refused to pay

The se matters. Compare:

• negar = to deny
  
  • Negó el error = she denied the mistake
• negarse a = to refuse to
  
  • Se negó a pagar = she refused to pay

So this is not just negar with an extra pronoun by accident; it is a different, very common construction.

Because after negarse a, Spanish normally uses a + infinitive.

So the pattern is:

• negarse a + infinitive
• se negó a pagar
• se negó a salir
• se negó a hablar

This works when the same person does both actions. Here, the customer is the one who refused, and she is also the one who would pay.

Because after hasta que, Spanish often uses the subjunctive when the action is still pending or not yet completed from the point of view of the main clause.

Here, at the moment she refused to pay, the correct bill had not been brought yet. It was something she was waiting for before paying.

So:

• se negó a pagar hasta que... sets up a condition that had to be met first
• that is why trajeran is in the subjunctive

A good way to think about it is: she refused to pay until they would bring the correct bill.

Because the main verb is in the past: se negó.

When the main verb is in a past tense and the subordinate clause needs the subjunctive, Spanish usually uses the imperfect subjunctive:

• present-time version: Se niega a pagar hasta que le traigan la cuenta correcta
• past-time version: Se negó a pagar hasta que le trajeran la cuenta correcta

So traigan would fit a present-time sentence, but trajeran matches the past narration here.

Le means to her here, and it refers to la clienta.

The idea is:

• trajeran la cuenta correcta a la clienta
• Spanish replaces a la clienta with le

So:

• le trajeran la cuenta correcta = they brought her the correct bill

This is an indirect object pronoun.

The subject is omitted because Spanish often leaves out subject pronouns when the context makes them clear.

Here, trajeran means they brought / were to bring, and the hidden they would be something like:

• the waiter
• the staff
• the restaurant employees

Spanish does this all the time. The verb ending and the context usually make the subject clear enough.

Because in Spanish, adjectives usually come after the noun.

So the normal order is:

• la cuenta correcta
• el libro interesante
• la respuesta exacta

Putting correcta before cuenta would sound unusual here and would give a more marked, literary, or emphatic effect. After the noun is the neutral, natural choice.

Also, la cuenta correcta clearly means the correct bill, as opposed to the wrong one.

Así que means so, therefore, or as a result.

It introduces the consequence of what came before:

• the customer refused to pay
• therefore, the boss ended up having to apologize

It is a very common connector in everyday Spanish and often sounds more natural and conversational than something more formal like por lo tanto.

Tuvo que is preterite, and here it presents the obligation as a specific completed event in the story.

• tuvo que disculparse = she had to apologize, and the sentence strongly suggests that she actually did

If you said tenía que disculparse, it would sound more like background information, an ongoing obligation, or something that was necessary in general, without focusing so much on the completed outcome.

So tuvo que fits better because the sentence is telling us what ended up happening.

Because disculparse means to apologize, while disculpar usually means to excuse or to forgive/excuse someone.

Compare:

• La jefa tuvo que disculparse = the boss had to apologize
• La clienta la disculpó = the customer excused/forgave her

So the reflexive form is the one used when the subject gives the apology.

Another common way to say this would be:

• tuvo que pedir disculpas

Al final means in the end or eventually.

It shows that this was the final outcome after the disagreement or delay. It adds a sense of resolution:

• first there was a problem
• then the customer refused to pay
• in the end, the boss had to apologize

It is very common in both speech and writing.