Officers established the woman to be Daniela Adamcova, 57, from Slovakia, following multiple tips from Croatia and abroad.
Ms Adamcova was reportedly a successful jewellery designer and artisan who ended up becoming homeless.
She had no recollection of who she was or where she came from when she was discovered bloodied and bruised on the rocky island in the Northern Adriatic on September 12.
The tip revealing her identity came from the US where she reportedly lived until 2015 before moving to Ireland, according to Croatian media.
Ms Adamcova was discovered on Krk sitting on a jagged rock close to the sea in a remote part of the island inhabited by bears.
She had no identification documents or a mobile phone, reports said.
Police released a photo showing a blonde woman with blue eyes and a bloodied face.
Croatian rescue services said she had spent a night at the seashore and was found “exhausted and with light injuries and disoriented”.
The case drew attention in Croatia and internationally.
The woman has since been transferred to a hospital in the town of Rijeka.
Police described her condition as “stable” and said social services will take over once she is released from hospital.