Quando a bateria dos fones sem fios acaba, continuo a estudar em silêncio.

Dos is a contraction of de + os (of + the).

• a bateria dos fones sem fios literally = the battery of the wireless headphones
• So dos expresses possession or relationship, like English “of the” or sometimes the possessive “the headphones’ battery”.

Structure:

• a bateria = the battery
• de + os fones sem fios = of the wireless headphones → dos fones sem fios

The subject of the verb is a bateria, which is singular, not os fones.

The structure is:

• Quando a bateria dos fones sem fios acaba, …
  
  • a bateria = grammatical subject (singular)
  • dos fones sem fios = a prepositional phrase describing bateria

Because bateria is singular, the verb must be:

• ela (a bateria) acaba → acaba (3rd person singular)

You cannot agree the verb with fones, because fones is not the subject here.