Learn Spanish (Latin America): El paciente acababa de dormirse en el quirófano. - The patient had just fallen asleep in the operating room.

This is a great example of a pronominal verb changing its meaning. While dormir simply means "to sleep", adding the pronoun to make it dormirse changes the focus to the beginning of the action. It means "to fall asleep" or "to go to sleep". Since the patient had just drifted off, we need dormirse.

Why is acabar in the imperfect tense (acababa) instead of the preterite (acabó)?

The phrase acabar de followed by an infinitive is a special formula. When you use it in the present tense (like acaba de), it means someone "has just" done something. When you put it in the imperfect tense (acababa de), it shifts to the past, meaning someone "had just" done something right before another past event. Using the preterite (acabó de) would literally mean "finished doing" something, losing that "had just" meaning.

The pronoun se is attached to the end of dormir. Could I put it somewhere else in the sentence?

Yes! Because acababa de dormirse acts as a single verb phrase, you have two choices for pronoun placement. You can attach it to the infinitive as shown here, or you can place it right before the conjugated verb: El paciente se acababa de dormir. Both are perfectly correct and mean the exact same thing.

What exactly is a quirófano? Can I just say "sala de operaciones"?