Everyone has a different approach to this game, and they say follow this, follow that. I do what works for me, and if I feel like a girl is a waste of time, I will drop it and leave as that. To help me get over her and not swell on the fact, I look at all of her negatives and things I don't find attractive, which helps me let go.
So since she counter-offered, people would say "That's okay -- it's only when she denies, she is a waste of time."
In the common court of Law, if someone does you wrong and owes you money, they can request a monthly payment and the judge must accept all offers if the defendant denies to pay the sum amount monthly. The defendant can only COUNTER-OFFER, which then that must be accepted. But when they deny the amount, the Judge says "You cannot deny the amount, only counter-offer -- so that concludes this court session, you pay them $50 monthly." You could have paid them $5 monthly if you would have said "It is in my best effort humanely possible to pay $5 a month." - in which the judge must accept.
The same goes for this kind of date -- if she denies but never says "Can we do next week?" - she is gone and not interested. If she is that excited about it, she will make it HER best effort to date you as soon as possible!
I say, continue to talk to other women (I know, you find someone so physically attractive and you find it hard to find someone else - but when you do, you look back and say "wow, this one is so much better, nicer, and attractive.") and when next week comes, pick a physical day and time and if she says she is busy, then it's time to let it go.
She will become like us men when women stop initiating contact first. She will worry, "Oh wow, he isn't messaging me anymore for dates. He isn't texting me anymore, which I loved. I must get him back so I'll worry, and send the first message."
Once you do that, she will know you aren't playing around and you won't play 'games' because you didn't let her play the same games on you. If you let her push you around, she will know you do the same thing and women don't like 'games'.
The reason why she didn't counter-offer with a definite time is because you either did not give a specific date and time.
Wait until next week - ask her with a physical date and time - and see what she says. If she says no, play nice and make it seem like it didn't affect you. If you go crazy, it will confirm her feelings with "Wow, this guy is a complete *******. He assumes I'm not interested, so he will just ignore me. Not the guy I want."
Play friends, and be like "Okay that's fine!" and end it there. OR, "Okay no problem." Don't make it seem like you're willing to accept the apology ever-so-quickly. But don't feel offended by her decision.
Play cool -- play like it's her loss, and know that it might not be your fault - she could have major trust issues, and is afraid to get close to someone. It could be because she might think you're after a relationship, it coming on too strong.
She could also be holding it out because she wants your desire - she wants to feel wanted, and the only way to do that is watch you come back over and over again, begging. Once you stop, she will want those feelings back.
Men do this with women - they will constantly pick fights, only to push the girl away and have her come back, fighting for you attention. Then the man feels powerful, and the relationship is fine, and the process repeats.
So play their game! I had a girl text message me a few days, every night and the night she didn't, I was over my head driving around and thinking. Do the same thing -- initiate conversation, ask for a date, and then stop. She will wonder why, and she will miss those nightly text messages. I'll get sh1t for telling you that, because the 'rule' is that we are supposed to have them come to us, since we are the prize. The alpha's are far from being 'alpha', because they listen to what other people say, and they follow out - only later trying to believe it themselves, that they are some type of esoteric piece of attraction.
Do what works for you - I've started to follow these 'rules' and man, I suck... I've had more success years ago with my own way.
So try whatever out - don't let her play you, and leave you on the sidelines - stop asking her for a date (ask her next week) after next week, and see what happens. She will come back! I've had women hold on to me for months and then we finally dated, casual sex, what have you. I stopped asking and they always wondered "Are we going to date?" "Am I going to see my boy?", etc.
If she can't hangout, she could be pushing you away to see if you'd come back.
She will think - "He asked me twice, he got my number. He is interested but I didn't know that! I have to ask him out and say yes."
You leave the mystery in the game when you pull away, and it will ultimately be the attraction she needs to become attached to you.
If you allow her to play the games, if you get pissed when she says no, it will just confirm her feelings that you too play all sorts of games and you're not worthy of bonding with her.