Power
Vlad Masters has a poor taste in women, sure, but at least he has charisma and good looks—as well as the ability to make his enemies burn.
And now, after eluding Daniel and his bratty friends with the stolen Infi-Map, the half-ghost has a world. A world of his own. He has power, not that it's a new possession. But he has a world of people—as well as the ethereal minions he discovered in this world's spirit realm—to devote themselves to him.
Is it unconditional love? Well, it is hard to define a love as unconditional if a lack of such reverence can lead to a snapped spine or loss of limbs.
Azula isn't moved by this, though. She isn't moved by this spirit-man's attempts to treat her as if she's the true treasure. As if she's beautiful. It's all rigmarole she's heard from traitors and suitors alike. No matter what she told those quacks in the institution in order to be broken and bruised in a place where she could at least bathe unsupervised, love is venom. She doesn't need it because it debilitates her the moment it invades her thoughts. Makes her weak.
Any advances or bursts of passion or nightly romps with this fire demon are for the benefit of her position. The princess now has control over everything: her brother, her nation, the Avatar. All she has to do is pretend, and Azula is exceptional at pretending. His hands are cold, but they carry promises of domination and the tyranny that she craves—that she needs to hold.
But even if their touches are devoid of any love or warmth, he has power. And he shares that power because all Vlad has always wanted is somebody to rule alongside him in utter compassion or depravity—it no longer matters which.
If this beautiful, sick woman one day attempts to overthrow his grasp on this world, then it is simply a matter of which style of murder is the slowest and most agonizing.
Author's note: This is simply a disjointed product of my utter frustration concerning the fact that I am the only person that thinks this ship could sail. I mean, really. This is post-Infinite Realms in Phantom-verse and post-series finale in Avatar-verse. Also, I assumed that Vlad could take Aang in a fight; I mean, I know he's the friggin' Avatar and all, but even with ultimate power, Plasmius can not only duplicate himself an innumerable amount of times—but he can evade any of Aang's more dire attacks by simply turning intangible. Also, I have no patience, so I just dumped Vlad and Azula in bed together. Good times abound, yanno?
As for how Vlad could control the world, I assumed that loyalty is a lax commodity in the Avatar world. The Dai Li switched over to Azula like that, and the Fire Nation is apparently completely supportive of Firelord Zuko, even those that fought for the Fire Nation, despite the fact that they saw him as a traitor. I mean, if this wasn't the case, Bryke wouldn't have gone with the whole "the Avatar world is totally at peace now and there is no more war" system. Also, if that's too fragile of a foundation, I'll just go with the Vlad Is Just That Awesome argument.
Also, Azula simply lived in the Fire Palace as an abject, bitter failure because she lost the motivation to control people after her life's work of grinding people into her shoes and making them fear her went down the commode in a humiliating fashion. Zuko tried to bond with his dear sister over the turtleduck pond, but the turtleducks didn't appreciate it.