How do I go about negotiating higher pay for a current contract?

I started my own consulting business about a year ago, and have only one client as I have another part time job. The client offered the position at a rate that is much lower than the going rate for the type of work I do. I’ve been in the job for a year, and now that I know more about the market, I feel that I should be making more. Ideally I would be making nearly double per hour, and that is what I would ask of a new client. I realize that is not realistic in this situation. I would be willing to work at $15—$20 extra per hour. How do I make that ask?