Kuhn made her professional stage debut in 1981 and her Broadway debut in the 1985 original production of the musical "The Mystery of Edwin Drood". Subsequent Broadway roles include Cosette in "Les Misérables" (1987), Florence Vassy in "Chess" (1988) and Amalia Balash in "She Loves Me" (1993). For all three, she received Tony Award nominations. She also received an Olivier Award nomination for her 1989 West End debut playing Maria/Futura in "Metropolis". Other musical roles include Betty Schaeffer in the 1993 US premiere production of "Sunset Boulevard" in Los Angeles, and her Obie Award winning role as Emmie in the 2001 Off-Broadway production of "Eli's Comin". She received a fourth Tony nomination in 2015 for her role as Helen Bechdel in the original Broadway production of "Fun Home", and a second Olivier nomination in 2020 for her role as Golde in a London revival of "Fiddler on the Roof".