Gonna put some distance between us for a couple days as well
This is fine. Don't go weeks however. As
@fastlife said you'll fall completely off the radar screen and you'll leave too much space for her to end up going out with another dude.
If she isn't responsive when you ask her out this next time...move along.
You have to go by facts. Fact is she didn't have time (or wouldn't make time) for you recently. That is fact. It doesn't matter why. So you are already giving her the benefit of the doubt by asking her out again AT ALL. If she cannot make time for you the second time, then that is another fact. In that case she has established not once but twice that you are not the priority. Take the hint and get the message. It doesn't matter why. Just NEXT! and move along.
Now. IF she accepts your second date and shows up (you must always go by what someone does...not what they say...) then you gauge her interest in person by how she responds to you. Simple. You go based on actions and facts. That is all that matters.
I was talking to my ex husband last evening about this very same thing. There is a hot young woman he took out for Valentine's Day (and a time before that or two). I asked him if he kissed her. He said no, that she didn't/wouldn't kiss him. I know him well and know he is smooth with girls and I also know he is a good kisser and good in bed. But this chick he took out was lukewarm about escalation. Perhaps he needs to be more overtly sexual, perhaps he is allowing himself to get used a bit. The girl in question is accustomed to getting men to take her out and pay her way and entertain her for little investment on her part. She is very petite and hot so she gets away with it. I pointed this out to my ex, and he sees it as well. He is debating whether or not he wants to take her dancing this weekend (and see if he can escalate with her)...but right now the facts indicate she sees him as more a placeholder or an entertainer than a lover. Not good. I think he ought to lone wolf it and see what he pulls rather than waste time on this chick, but he is a good dancer so going dancing may play to his strengths. He will have to figure all that out.
The only other thing I would add is just a comment that there do exist traditional type women who expect a man to take the masculine role and lead. These women won't chase a man, even if they REALLY like him. You cannot tell if you are interacting with such a woman unless you are willing to initiate contact. If she likes you she will be responsive, encouraging, and she will facilitate getting together. She WILL make it easy. Some guys here will simply say that is "girl game" and admittedly it certainly is, but it also gives the woman in question a way to gauge you as the man based on your actions. If you reach out...you are interested. If she responds in an encouraging/receptive way...she is interested.
But notice how this approach is based in fact and action rather than wishes and fantasy. Don't over think. Just be the man and do. Her response will give you the feedback you need to proceed or not proceed.