The Women's Hospital, Showell Green Lane, Sparkhill, Birmingham, England
Notes: This is a tentative identification: The Women's Hospital isn't too far away from where she was living in 1901 (when her occupation was also given as housemaid) and there isn't another likely single woman of the right name and age in Birmingham.
However, who's to say she hadn't moved elsewhere? After all, her sister Mary Agnes was working as a nurse at a Lunatic Asylum in Berkshire. But, there's no other unmarried Ellen or Nellie Wright born in Warwickshire or neighbouring counties between 1883 and 1887 (according to the 1911 census index on Ancestry) anywhere else in England or Wales.
Could she have married between 1901 and 1911? She was old enough... There were 4 marriages of an Ellen Wright in the Aston or Birmingham districts between 1901 and 1911, and 8 possible husbands. Searching the 1911 census of Birmingham for an Ellen of the right age and birthplace and any of the 8 possible surnames only throws up a single possibility: Ellen married to Albert George Stephens in 1908. Luckily, it was a marriage in a Parish Church and Ancestry has the Church of England Marriages for Birmingham up to 1937. This Ellen's father was William, so she's ruled out. I could look for marriages of an Ellen Wright anywhere in the country between 1901 and 1911 but there are 296 of them — far too many to investigate individually. And, given her social class, it's most likely she would have married in Birmingham, close to where she grew up.
Emigrated? Six of her siblings went to Canada between 1912 and 1926, so she could have been the trailblazer. But I've not found her on an outbound passenger list between 1901 and 1911, not as a single woman and not under one of her possible married names.
So for now this looks like the best identification I can make.3