After getting a quick conversation going, you then want to see what time she might be available during the week so that you can figure out a time to get together. The best way I’ve found to do this, without coming across as weak, is to say something like, “Hey, I’m a little busy today and tomorrow, but I have a few days off after that, so I’m sure we can figure out a day to get together.” Or maybe this: “Yeah, I am booked during the weekend, but I have the next few days off. What’s your schedule like? I’m thinking we get together ASAP :)”
My only remark: I’d personally not wish to hear that he is soooo busy and has been too busy to call me before (see above – oh, this might be the reason why I have not answered your texts 😉) but I understand the intention to be not romantic but “laid back”, if I’d walk in your shoes. But from a girl’s perspective: you want to be laid back… why again? Because you feel like an idiot if she says no? And then you can say “Oh I really was not interested and just called because I was bored”. (?) No, really I personally don’t get it. From my experience guys who really dare, expose and are courageous are always appreciated more than these “cool/busy guys” (girls know instinctively if you are REALLY busy or just making it up to seem more interesting 😉 – trust me on this, we have to care for babies – instinct is our job). So don’t be afraid, after all men want sex, OK got it, but girls want sex, too – so what’s the point. It’s OK to ask a date. If she says no she is not worth it since she does not SEE you – another girl will, don’t worry.
But as this is the point where she did not get back to you and you decided to call (good decision, he gets a courage point). So what about this: Come up (instead of saying that you have been so busy) with a plan. (Think of the mammoth theory) – We LOVE men with plans. You could suggest to do some cool cultural or sportive or whatever thing, maybe you say you have tickets for something and would love to go with her. I think you make yourself more interesting like this. Theater, a cool party, indoor-climbing, black light Mini-Golf…whatever. (If you have talked to her you might have understood if she is sportive or culturally interested.) Generally girls don’t want to date “idiots” (whatever this means as a definition) so if you suggest to go to a museum (there are cool museums as well) or to some event like a vernissage or photographic exhibition she might think: “Uh, he has interesting hobbies and he wants to share this with me" (on top you will have this unforgettable first date and lots of stuff to talk about automatically – just don’t play paintball with her or other aggressive stuff – and if you do at least don’t laugh if you hit her).