I also wondered how they successfully fooled Karlie's mother, but I'm guessing here that sadly the conflict between Karlie and her mother worked in their favour. Maybe the person pretending to be Karlie kept her phone calls short, didn't answer in detail, made the phone call in a noisy place, used a bad connection, pretended to have a cold etc etc. Karlie's mum knew that they were on bad terms, maybe she was just grateful for any contact at all.
I also think that most of the contact was by text, and they only called when they thought it was really necessary. I'm just speculating here, but lets say that the fraudsters sent a few texts early in 2009. "Hey mum, we're fine. Don't want to talk." Then they stop contacting mum, being aware of the family conflict and hoping it will just be accepted. Mum sends further texts and leaves voice mail, they go unanswered. In September 2009 mum goes to the police and then sends a message saying, "I just reported you and Khandalyce missing, you'd better call me back if you're okay." The fraudsters panic, because they know if the police go poking around looking for Karlie they might find out that Karlie and Khandalyce are murdered. They end up deciding to risk a very brief, terse phone call. It works, and this emboldens them to make a few other calls, as well as text messages etc to extort money. Sometime in 2011 the pretense of being Karlie becomes too difficult or not worthwhile for some reason. Maybe a few things happened, such as Karlie's mother refusing to send more money, Centrelink needing proof of something they can't provide, prison time for unrelated crimes, spooked by investigations into Angel's body, another victim etc. They stop using the phone, and Karlie's mother and extended family think that Karlie has cut herself off for good this time. All speculation but I do think that something happened in 2011 to (mostly) give up on the impersonation aspect of the fraud.