You're not on the moon yet.
So far I've done my best to inspire students merely by showing them that I am interested in their work. I believe that if I project this attitude towards them, then they'll become engaged in the session with me. I'm trying to turn this into a conversation between the student and I about not just their paper but their style of writing. People like to talk about what they're trying to accomplish. They like people to listen to what their ideas are. They really do like this to be about them. And so I ask them about these things. I may not even give them the technical help they need in a given session. However, the student was allowed to express their ideas and hopes to someone who is really listening. That's what keeps people coming back for help.
As for the students who truly don't care about writing, I'll do what I can for them. But in many cases, it is only them that can discover whether or not they're passionate about writing. It cannot be contrived. Perhaps receiving a good grade on their paper is what makes them become more interested in writing.
I like to reflect on my own experiences in the Writing Center when I came in for help. My first draft of my first 2010 paper was atrocious. I really didn't think there was any hope for me and I wanted to drop my class. So I thought I'd give the Writing Center a shot. My tutor (I don't remember his name) was so effective at showing me what my paper was lacking and how I could turn it all around. He begged me for what my main argument was, helped organize my main points, and taught me the importance of a good consistent conclusion. Seriously, these were things my 2010 teacher wasn't telling our class and people seemed afraid to even ask.
In the end, I think I'll have to judge how to deal with a student's dispositon on a person by person basis. Give them candy or something.