Introducing someone to anime?

Specifically, my target is my bestfriend who rarely watch western cartoon, and only saw glimpse of anime in action. I'm limited to 10 days. That's how long I'm staying over during my trip to NY. Obviously, I don't want to overwhelm her by throwing the whole anime spectrum at her face. My idea would to watch anime around her whenever I can, and let her discover 'the magic' in her own time. But that's too subtle isn't it... So what would be the best way of going about it?

First you want to narrow down a genre, like do they usually like action, romance or comedy? And if English is their primary language then it'll probably be easier the first time if it's dubbed. It all really comes down to catering to what they like to watch and finding that gateway anime that sucks them in. Without details about who you're showing it to, we could speculate all day about what them may or may not like.

Okay, maybe I wasn't phrasing it right. I've already have the anime that she'll probably like. (pretty sure) but we don't have a whole lot of time to watch a dozen of them. (probably only 3 or 4 at best) The trouble is the execution. What I need to is to get her to be interested enough within this short amount of time. (remember 10 days) You know how some anime doesn't get "good" until say 5 episodes in (or even 10 sometimes) due to build up. I don't want have someone lose interest before any of that just because they don't know how to appreciate the build-up while at the same time trying to be comfortable with the sub-reading, anime artstyle, and awesome voice acting. (ENGRISH!)

"Hey you should check out this series I'm watching" "really whats that?" "Its called ________ its an anime you should check it out, starts slow but gets good!"

Watch the best scenes and series on a large TV when she's around. If it's interesting enough, it'll probably catch her attention as long as she isn't super preoccupied with something else. I have admittedly done this on purpose to get some of my closer friends into anime... Like having the battle against Kirito and Gleam Eyes coming on when they're just getting home...

Just like any other introductions, it's up to the person if they care enough to invest anymore of their interest. Anime already has the stigma of being "childish" and brands anyone that watches it as immature regardless of what the genre or age group is. The only way to introduce her to it is to just show her an anime you like. From the way it sounds, you want to do more than just introduce it to her, you want her to like it but you can't force her to (not saying you are). If she likes it she likes it, if she doesn't she doesn't. I understand that people believe having a lot of things in common make having a relationship easier but that's not always true. Anyway, my suggestion is to get an anime you like and just make it casual, don't try to "ease" into it. Just tell her you like and you wanted to show her to see if she likes it too.

Personally I'd pick something that's very easy to understand and not super over the top. Something like a slice of life show or anime movie thats very engaging and could be enjoyed in one sitting. I remember showing my mom an anime movie back when I was in high school cause she wondered what all thes "Japanese cartoons" I was getting into were about. I showed her the Wings of Honeameese (definitely spelled that wrong XD) and she enjoyed it quite a bit and could see why I like the animation style so much.