By the look of the births and the 1881-1911 censuses, Edith, the daughter of the landscape painter Henry Earp and Susanna, never married, so the closest living relatives would have to have been descended from one of her siblings. In 1881, the family were:
Henry Earp Head M 50 Westminster, Middlesex, England
Susanna Earp Wife F 43 Brighton, Sussex, England
Edith M Earp Daughter 7 Brighton, Sussex, England
Henry E Earp Son M 3 Brighton, Sussex, England
Percy G Earp Son M 1 Brighton, Sussex, England
Your line of enquiry should therefore focus on those with Henry Edward Earp (born 1878) or Percy George Earp (born 1879) in their family trees.
Henry looks to have married Nelly Johnson in Steyning (thats Hove) in 1912
and Percy looks to have married Maimie Webley in Burton (on Trent?) in 1914
If you search of FreeBMD giving the surname Earp and the maiden name of the mother, you will find the first names and birth years of the nephews and nieces of your Edith.
Henry had at least one child born in Steyning and perhaps others in Ashby De la Zouch.
Percy's son and daughter were born in Wolverhampton.
The daughter (born in 1917) is being researched by a member of Genes Reunited
Neither Henry or Percy died in WW1.
Roy G