Eileen was born in London and was a student at the Guildhall School of Music and Drama. She made her first appearance in London as Jaquenetta in Love's Labour's Lost for Robert Atkins at the Open Air Theatre, Regent's Park. Seasons in repertory followed, including two years with the RSC at Stratford-upon-Avon. She went on to work at the Old Vic, where her roles included the Queen in Richard II, Miranda in The Tempest and Viola in Twelfth Night in 1962.

In the contemporary theatre, Eileen's credits include Semi Detached with Laurence Olivier, Exit the King with Alec Guinness, and The Restoration of Arnold Middleton. She won the 1965 Evening Standard Award for Best Actress for her performance as Childie in The Killing of Sister George and made her New York debut in this play. She returned to New York in 1967 to appear in The Promise. In 1968 she appeared in The Cocktail Party at Chichester, which subsequently transferred to the West End, and in Peter Gill's The Sleepers Den at the Royal Court Theatre Upstairs. Eileen won a Variety Club Award for her role as Elizabeth in Robert Bolt's Vivat! Vivat! Regina at the 1970 Chichester Festival and later at the Piccadilly Theatre, a performance she repeated in New York.

Other London theatre credits include the title roles in Suzanna Andler at the Aldwych Theatre, St Joan at the Old Vic and Medea at the Young Vic. She played Nell in Passion Play for the Royal Shakespeare Company at the Aldwych. For the National Theatre, Eileen played Hesione in John Schlesinger's acclaimed production of Heartbreak House and leading roles in productions of Cymbeline and Pinter's Mountain Language. She received an Olivier Award for Best Supporting Actress in Peter Hall's production of The Winter's Tale and won the Critics' Award for Hanna Jelkes in Tennessee Williams' The Night of the Iguana, directed by Richard Eyre at the Lyttelton Theatre. She played Gunhild Borkman opposite Paul Scofield and Vanessa Redgrave in the hugely successful John Gabriel Borkman - again directed by Richard Eyre.

In 1989 Eileen received great critical acclaim when she appeared as Virginia Woolf in her one-woman show A Room of One's Own at the Lamb's Theatre in New York, where she received the Drama Desk Award for Best Solo Performance and a special Citation in a practically unanimous vote from the New York Drama Critics' Circle. Eileen then took the show on a short nationwide tour of the USA. She also recreated the role for Thames Television in a version filmed on location at Girton College, Cambridge, the venue of Mrs Woolf's original lecture.

Most recently she revived the play at the Hampstead Theatre. In 1992 Eileen premiered her own play, Vita and Virginia, at the Chichester Festival Theatre - as Virginia Woolf opposite Penelope Wilton's Vita Sackville-West - the show played a season at the Ambassadors Theatre. She then played it at the Union Square Theatre in New York to great acclaim with Vanessa Redgrave, and in 1995 received her fourth Tony Award nomination for Indiscretions with Kathleen Turner. She appeared in Edward Albee's A Delicate Balance with Maggie Smith at the Haymarket, directed by Anthony Page, for which she won the Evening Standard Award and played the Woman in the RSC's The Unexpected Man with Michael Gambon for which she won an Olivier Award for Best Actress. Eileen went on to enjoy success with this role on Broadway with Alan Bates.

In 2003 Eileen appeared with Corin Redgrave in Honour by Joanna Murray-Smith at the National Theatre, directed by Roger Michell, for which she won the Olivier Award for Best Actress. She appeared on Broadway in William Nicholson's Retreat from Moscow which was her fourth Tony nomination. In 2005 Eileen appeared as Meg in Harold Pinter's play The Birthday Party at the Duchess Theatre. 2006 saw Eileen play Sister Aloysius in Doubt, which enjoyed a successful run on Broadway at the Walter Kerr Theatre, directed by Doug Hughes. In 2007 Eileen appeared at the Almeida Theatre as Bridget in There Came a Gypsy Riding by Frank McGuinness, directed by Michael Attenborough.

Her film credits include: The Dresser, Equus, Jack and Sarah, Wolf, Gosford Park, Cold Mountain and Vanity Fair and, in 2007, Evening with Vanessa Redgrave and Meryl Streep.

Her television credits are many, the latest this year being Waking the Dead, Ballet Shoes and Miss Jenkins in Cranford for which she won a Bafta.

Her writing credits, apart from Vita and Virginia, include Mrs Dalloway for which she won the Evening Standard Award. She also co-created with Jean Marsh Upstairs Downstairs and The House of Elliot.