The Office of the First Lady (OFL) is the staff accountable to the first lady of the United States. The office and its responsibilities, while not constitutionally mandated, have grown as the role of the first lady has grown and formalized through the history of the United States. The Office of the First Lady is an entity of the White House Office, part of the Executive Office of the President. It is located in the East Wing.

The first lady has her own staff that totals 11 in number.

Though the persona, activities, and initiatives of the first lady have always been significant to the history of the United States, the first first lady to hire federally funded staff was Edith Roosevelt, who hired Belle Hagner as the first White House social secretary on October 2, 1901. Eleanor Roosevelt became the first first lady to expand the office beyond social and administrative secretaries by hiring Malvina Thompson as her personal secretary and Jackie Kennedy was the first to employ a press secretary.

Under Rosalynn Carter, the first lady's staff became known as the Office of the First Lady. She organized the office into four major departments: projects and community liaison, press and research, schedule and advance, and social and personal; and was the first to add a chief of staff. She was also the first to move her own work office into the East Wing. Though the role of the office has grown over the years, it primarily supports the first lady in promoting the agenda and campaigns of the president. Further to that, it provides support for the agenda of the first lady, who chooses causes and initiatives to campaign for during their time at the White House.

https://en.wikipedia.org/wiki/Office_of_the_First_Lady_of_the_United_States

No such thing as "office of first Lady"