Now that you've written it you might as well send it to her. The good news is that if it tanks it you won't have the awkwardness of seeing her around again. The worst case scenario here is that you get a letter of disinterest back the worst being if she harps on about her disinterest. Second worst thing not hearing back at all.
I personally think if you don't send it time will clock on as it always does and you will always have an occasional sad moment that you didn't send it, possibly worse over time, 10, 15, 20 plus years from now? How would you feel? So yes I would just send it, don't count on too much or a positive Hollywood ending. Treat it as at least finding out, you might get lucky but equally you might not, but the upside in that is at least you know and you learn by experience.
Personally I would never send a letter to a girl who I saw regularly. It's a old romance cliché/Hollywood Romance Movie cliché that often isn't a great move in the real world. It's kind of bad that Hollywood put that across as the way to go. It can cause great embarrassment and awkwardness moreso if you regularly see the person. While some girls like it I think probably only from guys they are already attracted too. So better to try a more low key approach usually, get to know the person so they are familiar to you so an approach doesn't come over as weird then hang out bait like, 'I want to go and see this film but all the people I know aren't available'. I think putting it all on the line is often a needless high stakes low reward method, unless you're a pretty boy of course but then it's rarely needed anyway.