The 33-year-old, named locally as Shani Cockrell, was found with critical injuries at a house in Galway Road in the early hours of Friday 27th December.
She was taken to hospital where she remains in a serious but stable condition, which has shown further improvement in the last few days.
The body of a man, Paul White, was also discovered at the property.
A Home Office post-mortem examination found the 37-year-old died as a result of a gunshot injury. His death is not being treated as suspicious and a file is being prepared for the coroner.
Officers are satisfied the woman was the victim and are not seeking anyone else in connection with the inquiry.
Two children found in the house were unharmed.
Detective inspector Ash Wilson is leading the investigation. He said: “This incident happened over the Christmas period at a house in which two very young children slept.”
“It has devastated two families, who now face a new year having to try to come to terms with what has happened.”