JERSEY CITY, N.J.—Katie Brennan spent more than a year trying to get authorities to take action against the man she accuses of sexually assaulting her. Finally, she emailed New Jersey Gov. Phil Murphy.

Ms. Brennan, chief of staff at the state’s housing agency and a former volunteer for the governor’s campaign, first called the police in April 2017, one day, she said, after a campaign staffer allegedly forced himself on her. After months of investigation, the county prosecutor’s office declined to file charges against her alleged attacker, saying it didn’t think a jury would convict him. She turned to state law enforcement and high-ranking members of the Murphy administration. Nothing changed.

Finally, the 31-year-old Jersey City resident, who said she turned down a $15,000 settlement offer from her alleged assailant if she signed a nondisclosure agreement, emailed Mr. Murphy and first lady Tammy Murphy directly.

She didn’t explicitly mention the alleged assault in her June 2018 email, but asked to speak with the governor or his wife about a “sensitive matter” that had happened during the campaign. The governor responded within the hour and said he was looping in staff to arrange a time.

“Hang in,” the governor wrote to Ms. Brennan in the email, which was reviewed by The Wall Street Journal. “We are on it.”