sorry all, you can just tl;dr if you want, wrote too much.
Go for it. Maybe. Depending on how you broke up. I used to date this one chick for a while and then we broke up because I was too young, went out with other girls at the same time and lied a ton while forgetting about the dates and other crap. She got tired of me lying and we got in a fight. Both being proud we never called each other until I called back like 5 years later as she was one of the best girls I ever dated and there were a ton of pleasant moments with her. I went over with just a nice bottle of wine, we sat by her house and went over some memories while she said a lot of stuff like "I still sleep with a teddy bear you gave me on that one date bla bla, think about you all the time bla bla" and it sort of went for a good start. The problem was that I came back to Uzbekistan for summer while I was already living in US. Then I got together with my old friends and wend back to partying, forgetting to call he back again and screwing it up completely. I called to apologize one day and her mom told me to never call her daughter again. That was it. Not that I care now, but it could have started all over.

Basically what I am saying its your call and do what you feel you should do, but don't send her crap. Go there 1 day before her birthday and give her a gift in person. Don't apologize, give some kind of "friendly" card with a nice bottle of wine and something meaningless, useful but memorable that she can put on a shelf or use day to day. Don't spend too much, average present such as a nice set of keyboard and mouse or toy or something of you choice since you know her pretty well (think of what she likes, hobbies, etc). You can also get her a gift card to a nice restaurant and depending on how the conversation goes, give or not give it to her. If she will look like she is missing the times, give it to her and offer to go out for her birthday or some time after. If she will act friendly, give it to her and say that depending on her choice she can go out with friends/family and that you would like to join, but it is her choice. If she will act like a bitch, keep it and go there by yourself, get drunk, forget about the whole thing and move on.
Just don't get to "being friends". If she would act like she would want to "be friends with nothing personal" for too long, forget about her and move on.
And remember to be confident, don't be humble and act like you don't really care at this moment.
Or take an easy way out, and take the advice of the rest here. There are a ton of girls out there and one is always better than the other...