I don't think it's bad to back off and see how she reacts. Heaven knows, you don't want to waste any more money dating a girl who's not interested in a relationship with you.
Anyway, I'm a girl and even I don't know if there's a recommended amount of time before the girl takes the initiative. I just know that if I'm keen, he will very much know how I feel. By 5/6 dates, I'd well and truly be initiating dates and conversation.
Tell me, has she had much physical contact with you? I'd also add the amount (or lack) of physical contact into the equation to help determine how keen she is. At the very least, I'd be expecting to see some kisses.
Never regret anything that has happened in your life. It cannot be changed, forgotten or undone. So, take it as a lesson learned and move on.