In September 2013, Peake was appointed an Associate Artist of the Royal Exchange Theatre in Manchester. Her association with the theatre began in childhood and she was a member of the youth theatre. Major productions in which she has performed include The Children's Hour in 2008, for which she won a MEN Award, and Miss Julie in 2012 for which she won a Manchester Theatre Award. All of her performances at the Royal Exchange have been directed by Sarah Frankcom with whom she also collaborated on The Masque of Anarchy in 2012 for the Manchester International Festival. Building on this work, in September 2014 Frankcom went on to direct her as the title character in a radical re-imagining of Hamlet. The demand for tickets was so great that the production was extended for a week, having been "the theatre's fastest-selling show in a decade". The Guardian said of her performance: "Peake’s delicate ferocity, her particular mixture of concentration and lightness, ensure that you want to follow her whenever she appears". A year later she appeared in Frankcom's production of The Skriker, as "Caryl Churchill's shape-shifting, doom-wreaking fairy". The Guardian's Lyn Gardner listed the production in her top ten British plays of the year. In 2016, Peake resumed her partnership with Royal Exchange Artistic Director, Sarah Frankcom, to star as Blanche Dubois in Tennessee Williams' A Streetcar Named Desire. Peake's performance in the role garnered critical acclaim with The Guardian describing her performance as "exquisite" and "breathtaking."