I’m trying to help here, really. Now, if I wanted to draw up some conversation; an example of how not to get the girl, the above would be it.
And this:" So then it’s Saturday. She says she might be going out with her two friends, but they don’t have any definite plans yet. I inform her that my friend Paul’s fraternity is having a party and they can feel free to come over if they can’t find anything else to do."
So you’re the amusement of last resort?
You show no confidence. You make it plain that you’re available for whatever purpose she wants, at whatever time she wants. You make it plain that if your presence is an inconvenience to her, you’ll drift away for a while. This won’t work out for you. Girls like confidence, not wishy-washy BS. So you’re not confident? Then fake it!
You are going to have to take some initiative here. You are going to have to put some pressure her. Ask her out and make it plain that it’s a date, not just “hanging out”. Don’t say, “Lets go out sometime, somewhere.” Approach it like this: " I want to take you out to dinner Saturday night. I’ll pick you up at Seven O’clock. After dinner we’ll go to the [pool room/art show/dance/ submarine races/what the hell ever]. No way, no how, let the word FRIEND get into this conversation.
Don’t do this online. Next time you are talking online, type in that you’re tired of the computer and you’ll be over to see her in a few minutes. If she says she’s depressed again, tell her she needs some company, and you’re just the man for the job. Tell her you’ll be right there and sign off. Then go!
Ask her out face-to-face. If she blows you off, you’ll at least have some closure on the issue and you’ll know to move on.
Who am I to be giving all this advice? Nobody special, but I’ve never had a problem getting a girl to go out with me. Keeping one might be a different matter, but that’s not the issue here. There have been a number of girlfriends in my life and two wives, so I’ve got some t-shirts.