That's the Conquered Earth series (2012-2014) by J. Barton Mitchell.
Earth has been conquered by an alien race known as the Assembly. The human adult population is gone, having succumbed to the Tone---a powerful, telepathic super-signal broadcast across the planet that reduces them to a state of complete subservience. But the Tone has one critical flaw. It only affects the population once they reach their early twenties, which means that there is one group left to resist: Children.
Holt Hawkins is a bounty hunter, and his current target is Mira Toombs, an infamous treasure seeker with a price on her head. It’s not long before Holt bags his prey, but their instant connection isn’t something he bargained for. Neither is the Assembly ship that crash-lands near them shortly after. Venturing inside, Holt finds a young girl who remembers nothing except her name: Zoey.
As the three make their way to the cavernous metropolis of Midnight City, they encounter young freedom fighters, mutants, otherworldly artifacts, pirates, feuding alien armies, and the amazing powers that Zoey is beginning to exhibit. Powers that suggest she, as impossible as it seems, may just be the key to stopping the Assembly once and for all.
All adults above a certain age succumb to some static they hear and their eyes turn fully black and walk towards these ‘towers’
At the very beginning of the book, Holt (the main character) encounters a group of kids in their late teen years. They have black veins in their eyes.
They were younger than Holt, but not by much. Seventeen or eighteen he guessed, looking at the black veinlike growths crawling through their eyes, the telltale sign of the Tone. It had a firm hold on them now; it meant their time was running out.
The "towers", however, are actually alien ships. They're called "Presidium". (Presidia?)
When a survivor finally Succumbed, he began a long, slow, zombielike walk to the nearest Presidium, the massive Assembly base ships that had come roaring out of the sky eight years ago, impaling themselves into the hearts of the world’s great cities like daggers.
There are adults who are immune to the call, they are called ‘immunes’ I think. The male protagonist is one such person.
Nope! Apparently, the name is "Heedless". And yeah, indeed, Holt is a Heedless.
“Hey, this guy’s a Heedless, look at his eyes!” one of them pointed out bitterly. They were right - Holt was Heedless. One of the rare few on the planet the Tone didn’t affect. His eyes were perfectly clear; there were no signs of the crawling black tendrils. It was the only reason Holt had made it to twenty years of age.
The girl is able to cure people suffering from the static.
Holt had mentioned this, the light, but it was all new to Mira. She’d missed Zoey doing her thing at Midnight City. She had been lost at the time, and the little girl had saved her. Another of her powers, the most important one. Zoey could stop the Tone. Block it somehow, make you immune to it. It was still mind-boggling to think about.
The protagonist built and hid an artefact which would accelerate the rate at which a person succumbs to the static and can even affect an immune person.
Mira, the second main character, had built an artefact that was supposed to slow down the Tone, but failed big time.
Holt had only seen the artifact twice since they’d left Midnight City. Mira kept it deep in her pack, as far away from her as possible. She hated it. It repulsed her, and for good reason.
It was the ugly result of an obsession with forging a combination that could slow down the Tone but it had all gone wrong. The combination didn’t slow down the Tone, it accelerated it. Made it so that anyone, Heedless or otherwise, would Succumb in a matter of seconds. Making it had cost her everything—her life in Midnight City, her freedom, whatever future she might have had.
In the world exists aliens and artefacts, which come from an area that is separated into different tiers (depending on how close the ring is to the centre of the ‘shattered tower’), different items matched together can create different metaphysical properties.
The characters travel to destroy the above artifact. There is a mention of components the property of which are related to the "rings" they were originally from.
She was bringing it into the Strange Lands to destroy it, and Holt didn’t blame her.
“You okay?” Holt asked.
Mira stared at it a moment more, then stuffed it down into her pack. “Yeah.”
“You can destroy it at this Crossroads place?”
“It’s not that easy.” Mira’s voice was bitter. “To destroy an artifact, you have to be in the ring where it was created. If it’s a combination, you have to be in the ring of its most powerful component.”
“So what ring is that, then?” Holt asked.
The second book is named after a city in the trilogy where hunters gather and trade black market ‘artifacts’.
The second book is titled The Severed Tower. The severed tower is located in the Strange Lands, where components for artefacts come from. You might also want to notice that the first book is called Midnight City, and the character Mira comes from said city; she's able to craft a lot of artifacts.
A group of rebel aliens split from the main faction and aid the girl (and the main characters) in their quest.
Indeed, some Assembly (the alien race) individuals made some kind of a splinter cell and allied with the protagonists:
“Can I say something here?” Holt interjected, and everyone looked at him in surprise. Clearly, they’d forgotten he was on board. “We’ve made some … strange alliances, it’s true, but those Assembly you’re talking about are different from the others. They’re fighting their own kind. Now maybe that’s not something you particularly care about, and I don’t blame you, but at the very least you need to recognize that things are changing, and you ought to be concerned about a lot more than just arming your ships. The Strange Lands are gone. The Assembly are fighting each other, and whatever their agenda is, it’s reaching its end.”
Found with the Google query
book immune black eyes tower site:goodreads.com; the first result is the second book in the series, The Severed Tower. All quotes courtesy of Google Books.