Jenny trained at the Bristol Old Vic Theatre School, graduating in 1979, and first made her name in the film Local Hero (1983), playing the zany Scottish lover of an American oil baron. She is now busy on screen again in Judge John Deed, and in the intervening years was mostly busy working in theatre. In 2002, in-between series of Deed, she took on the title role in a West End revival of Somerset Maugham's comedy The Constant Wife. Other recent theatre includes Brief Encounter and The Odd Couple.
Jenny was married to the Indian actor Madhav Sharma, but that ended in divorce. She previously lived for several years with the film director Michael Winner. She now lives in North London with her partner, the actor, producer and theatre impresario Bill Kenwright. She is a vegetarian, an animal rights campaigner, and a fan of Everton Football Club, where Bill is a director. Her career was celebrated by the British television show This Is Your Life in March, 2002.