It pains me to say this, because I really enjoy the fruits of your misled labor, but buying girls drinks is the worst icebreaker you can deploy. In fact, I can confidently say the woman above is not looking at the dude panting about how suave he is. She may be smiling, but she's thinking, "This poor soul just wasted $16 on a cocktail and I have no intention of even having a conversation with him."
Sending her a drink is the most unoriginal, ineffective way to spark a conversation because it immediately indicates you're insecure. And, as I'm sure you're aware, even the slightest hint of insecurity gives off an invisible, anti-pheromone scent that unwittingly repels her. It reads as fear. You need to muster the confidence to make the first approach. No gimmicks. Just you, being you.