"How to win my girlfriend back?"

It sounds like you're right and it needs to be big in some way since you are saying you screwed up real bad.  However, defining "big" in her eyes may be difficult.  For some women, something very expensive might do it.  Remember Coby Bryant's wife with the four carat diamond?  Other women (I am one) would feel like I was just being "bought off" and that wouldn't work for me.  You know this woman so can probably tell if she would like the big financial output or not.

The thing that a lot of women would like is something that took a lot of thought and effort on your part and was individualized to something she likes or wants.   Does she like daffodils?  Go on the internet and find everything in daffodils you can - dishtowels, earrings, kitchen cutting board (I have one!) - you get the point.

Does she like something other people would see - for example, skywriting, or a big surprise party of some kind, so her family and friends will all see how much you love her? 

Or does she value privacy more, and would like a more intimate setting, where maybe you cook her a nice dinner - her favorite meal, play her favorite music, etc.  If you cheated on her (obviously, I don't know), how about something a little silly like tying a bow onto "Mr. Happy" and promising her she is the only one who can open that package?  (as long as you mean it; certainly don't make any false promises).  If it was something involving a problem behavior or addiction, promise to get help and make a start and show her what you've done and follow through.

Mostly, be honest with her.  Tell her exactly how you feel.  Tell her, if true, how badly you feel about what happened, that it will never happen again because you don't ever want to hurt her again and don't every want to risk losing her again.  If you want to spend the rest of your life with her, tell her so.  Tell her what she means to you.  Do you think of her right before you go to sleep and first thing when you wake up?  Tell her so (leave out the sexual desires part at this point). 

To me, the most romantic things I tell my husband are that I love SLEEPING naked up against his fuzzy chest, that I enjoy the most mundane tasks, like chopping carrots, or grocery shopping when we're together, that I can't imagine life without him.  Do you love the smell of her hair?  Does her laugh always make you smile?  Do you think of her as your soul mate?  Pour your heart out.  You are risking having it stomped on, but it's better than wondering what might have been if you really love this woman.  And if you really love her,  chances are she'll know it and know that kind of thing doesn't come along too often - never for some people - and not want to throw it away.

When you compliment her, never compare her to other women (my husband did this at first and even though he was being complimentary, I really didn't want to be constantly reminded of the other women he'd been with).  Just keep it on her.  This may not even be an issue with you, I just thought I'd throw it in, just in case. 

Also, if you two patch this up, and I hope you do as it sounds like you really love her, make sure this gets resolved and put a closure to it.  You don't want this being "held over you" forever.  She needs to forgive you.  If what you did was really bad, it may require couples counseling.  But it's worth it.  She doesn't need to keep the anger and you don't need to keep feeling her anger.  For your relationship not to be poisoned, it needs to be put in the past and the two of you need to move on.

If she isn't seeing you because she's still angry, bombard her with flowers, candy, singing telegrams, anything and everything.  Fill her home and office (if applicable) with flowers with love notes attached.  Does she have e-mail?  Send her a copy of this question you put on Askville!

Pull out all the stops!

Good luck!  Sounds like you are willing to do a lot to get back together.  I hope she can forgive what you did and appreciate your efforts to patch things up.
Sources: personal opinion and experience