I do agree with GLYC that, generally, if you get shot down once it probably means the person isn't interested. Generally speaking, you aren't going to change their minds.
...But that doesn't necessarily mean that is always the case. So, given your circumstances, I think my advice would be to go for it. Even if only because this way at least you'll know if she's just not interested in you, or if maybe something about the way you asked before just put her off. Maybe she'll just decline again, but at least then you will know.
Because, if she was interested then she PROBABLY would have said yes then... but it certainly could be possible something about the way you asked caught her off guard and she just kind of, on a gut-reaction, said no. You never know.
As for how, to be honest, I don't think the how matters all that much. Within reason, of course. If somebody is interested, they will be no matter how you ask, and if they aren't interested they won't be no matter how you ask. Again, within reason of course. If you said something to her like "Hey, want to see my collection of lamps made out of my human victims?" That might be different. LOL! I mean, at least that line never worked for me, anyway.
I'm sort of on the fence about whether your new approach involves mentioning/apologizing for the last time. I sort of lean towards thinking you just shouldn't even bother to bring it up... but I'm not 100% sure. Maybe I'm wrong on that. So, I will at least say this for sure. If you DO decide you'd feel better mentioning it... I would suggest doing it sort of casually/almost like a half-joke. Don't make a big deal out of it like you are really apologizing for doing something wrong. Just laugh it off almost like "Wow, I was such a goober back then." I think making too much of a big deal about it could just have the negative side effect of reminding her about whatever it may be that caused her to say no, if indeed it had anything to do with how you asked.
Good luck.