I had a relationship just like you described. My ex-girlfriend could not be bothered or motivated to find out what she wanted to do, and every attempt at getting her to find out would either end in a fight and a flat refusal or a promise to look into it, only for her to invariably break that promise...
I understand your frustration. My ex's problems ran a little deeper than not knowing what she wanted to do in the future, but the core problem was the same, no matter what I or anyone did, she could not be galvanised to do anything at all. And it's a horrible feeling, because I cared for her a lot. You think she'll turn it around, or you'll be able to make her do something about her future and then... nothing happens, nothing changes. And I fully believe you when you say you tried to encourage her and I fully believe you care for her, else you wouldn't start getting resentful and frustrated.
I can only tell you what happened to me - my relationship only lasted a month and a half from the moment I realised I was getting frustrated at her lack of plan or action towards her future and her other problems. But I looked back and realised that I had been frustrated for months beforehand, probably refusing to accept it. You probably have come to a similar conclusion or will soon. And then I came to the conclusion that people like my ex don't enjoy being like that, but are also convinced that having an aimless, purposeless life is better than the alternative of trying to find out what to do with their lifes. I believe that no one but themselves will make them change. You can try and aid them, but the responsibility lies squarely with them and if they don't want to, there's absolutely nothing you can do about it. I also noticed that these people bring you down, unknowingly and unwittingly, but inactivity and aimlessness is contagious and you will see in a few months that if you were to pursue a future with this girl, she was going to not get anywhere in life and stop you from going where you want to go in life. You'll also see that you weren't doing her any favours and were also unknowingly and unwittingly facilitating her not doing anything about her future.
If you had to ask this question, it's because you've already answered it yourself and you just either don't know it yet or you refuse to accept the answer. You are fighting a losing battle. Your relationship is on borrowed time and the longer you avoid the inevitable, the more you'll hurt her - and yourself. I'm very sorry. Life sucks like that sometimes.