Is there expiation for this act, will Allah accept the repentance of its doer, and is it possible for them to be afflicted with an illness as a result?

Your friend did well by taking the initiative to repent from committing the indecency, and "the one who repents from sin is like one who committed no sin." No specific expiation is required of him, but he must increase good deeds, avoid everything that might lead him back to indecency, and ask Allah for well-being. Your friend erred by informing you and his wife of what happened. What was incumbent upon him was to conceal his deed. He must repent from revealing his secret and not repeat such an act.