How can my husband purify his money after learning that designing advertisements for girls with their hair uncovered is unlawful, especially since he used a large part of it to establish his office (his source of livelihood) and does not know how much that part was?

Working in advertisement design varies in its ruling depending on the advertisement. If it includes a picture of an immodestly dressed woman, then it is unlawful (haram), and the earnings from it are unlawful because they are in exchange for an unlawful benefit. The perpetrator must repent and dispose of the money earned from it by spending it on charitable causes and for the benefit of Muslims. If he does not know the exact amount, he should donate what he believes to be most likely the amount. The disposal of money refers to giving away its equivalent value, not the specific money itself. So, if he invested it and made a profit, the profit is lawful for him, and he is only required to give away the unlawful money.