First I would never ask for a raise if the project has a definite scope. If the project has a definite scope, you give the client notice that after the scope is complete, further work will be at a higher rate.
Secondly "when" is a question that cannot be answered by someone outside. That's a business decision and it has to do with your relationship with your client, to what extent demanding a higher rate risks your contract, etc. These are business decisions you have to make based on your local knowledge of exactly how things are.
Third, a major variable here is how the dynamics are set up. If this is a contract for hire chances are they originally set the rate. If you originally set the rate you are in a better position than if they did. This is also a proxy measurement for whether you are approaching yourself as a service vendor or just a pseudo-employee. At least in theory you are a service vendor and free agent though.
My recommendation would be to try to set up a meeting. In the meeting you want to explain that the project hasn't gone the way either of you planned, that you would be interested in full-time employment still if that is on the table, but otherwise, you would like a pay increase, perhaps effective in a month or two. If it were me because I am comfortable working on my own, I would offer to transition my role into that of a consultant so that they could keep me (on my terms!) for as long as they'd like without tax headaches. Whether you are comfortable with that given your interactions with this employer is another matter so that has to be your call.