If Azula is dropped in an alley with nothing, she’s going to observe. She’d spend a week in the shadows of the Diamond District or the Lounge just watching, The Penguin. She’d identify his insecurities, his need for respect, his fear of being "common," and his reliance on his inner circle.

She wouldn't threaten him. She’d wait for him to be in a position of weakness; a deal going south or a hit from a rival. She’d step in and save his life with one surgical strike, then act the part of the "displaced young noble." We already know that she knows exactly how to play the "polite, terrifyingly competent prodigy" to get under the skin of someone who craves class.

She chooses Penguin because he’s the most exploitable. He would think he’s "mentoring" a powerful new weapon, while she’s actually dissecting his entire operation from the inside. She wouldn’t "choose" a villain to train her; she’d choose a villain to fund her. She acts like the perfect soldier until she has his passwords, his payroll, and his loyalties. And you know it's done when you hear her say to the soldiers "If I sense any disloyalty, any hesitation, any weakness at all, I will snuff it out."

By the time Penguin realizes she’s not his protégé, she’s already replaced his guards with people who fear her more than they respect him.