Kyle, I'll begin by saying I don't know you. I don't know your age, where you live, or really, anything about you except what is posted in this thread. So, I'm working from a clean slate. Vaskidmark is absolutely correct. Others here have said essentially the same thing, but toned down a bit to (presumably) be a bit easier on you. Facts are facts. You are risking this girls life(either physically or emotionally) for your own gratification.
You say her family "isn't the type" to engage in an honor killing. While I agree that honor killings amongst muslims are less common than the media might have us believe, you don't really know these people, do you? Other than through the rosy red glasses worn by their daughter, who seemingly is also deep in thrall to her hormones. People have a tendency to go off the deep end when they find out their little girl has been "despoiled". Honor killings, while perhaps more rare than amongst muslim families, happen here in the US in Christian households, except they aren't called that. It usually happens when drunken daddies find out their little girl is "a whore, and has been fooling around with that Johnson kid." Daddy loses control, and beats his daughter to death.
So let's not pretend you actually know these people, and let's not pretend you have deep, meaningful feelings for her. If you did, you wouldn't be asking us for advice on how to have a few more days with a girl who refuses to even be seen in public with you. Honestly, let's put this in perspective, she likes you, I am sure, or she wouldn't be risking community condemnation/honor killings/whatever to spend time with you. But neither does she have especially deep feelings about the affair, since she quite clearly has no intention of breaking with her family over it. Or perhaps, she is a pragmatist, and simply trying to have whatever fun she is able to while she can. Either way, the two of you have no future. None. Zilch.
Life is not fair. In fact, often it is the farthest thing from fair. But, if you work hard, man up and throw away all that malarky about getting your "needs" met in a meaningful way(much like I was taught in school), by the time you are 30 or so, you might meet someone special, whose circumstances are more conducive to a lasting relationship.
Or, you might end up going through life alone. But as Sindawe said, there are worse things.
Life is hard. It's harder when you're stupid. Let's hope you aren't, and you do the right thing by this girl(and yourself).