In "Jessica Jones", Jessica wants to stop Kilgrave without killing him. Because Kilgrave needs to talk to use his powers, why doesn't she stop him from talking with a brace. This has been done in other TV Shows in the Marvel Cinematic Universe, like in Agent Carter and in Agents of Shield
Because "de-powering" Kilgrave was never the goal.
For most of the season, Jessica wanted to prove Kilgrave and his powers were real in the hope (pun not intended) of exonerating Hope Shlottman.
Jeri: There's nothing you can do for this girl.
Jessica: I can prove that Kilgrave exists.
Jeri: If there really was a man who could influence people like that, I would hire him to do all my jury selection.
Jessica: [shouting] God damn it! This is not a joke.
Jeri: How? How will you prove that this mind controller is real?
Jessica: I'm resourceful. But if I can convince you, will you represent her? [sighs] I'll owe you a favor.
Jeri: Excuse me?
Jessica: If you do this, I'll owe you a favor.
Jeri: All right, convince me.
Jessica Jones S1E2 AKA Crush Syndrome
When Will Simpson later suggests killing him, she makes it very clear she's not interested in a lethal solution (yet).
Jessica: But he's not gonna let me sneak up on him. So I need to deliver the sufentanil with a long-range dart gun.
Simpson: Won't work if he's on the move. A long-range sniper attack takes setup. Your subject needs to be in a predetermined location, preferably removed from civilians. [...] It'll take a pistol. Up close and personal. He'll be dead before he hits the ground.
Jessica: I need him alive, Rambo. I'll keep him doped up, I just need a place to stash him.
Simpson: A bullet to the head is more effective.
Jessica: Not for the girl who's sitting in prison, taking the blame for his murders. Without proof that he exists, she's done. And that's on me. So this is my call. My call.
Jessica Jones S1E5 AKA The Sandwich Saved Me
Not to mention the times later in the season where Simpson tries to kill Kilgrave and she chooses to stop him.
As to why they didn't use a brace for the short-term depowering needed to capture him, I can think of two reasons. First, it's much easier to move someone around and ensure they won't escape when they're completely knocked out. Second, it wasn't until late in the season that we learned exactly how Kilgrave's mind control worked. At the beginning, all Jessica knew for sure was that "time and distance" helped it wear off.
Jessica: His control, whatever it is, it wears off. But it takes time and distance, so we're both getting out of here.
Jessica Jones S1E1 AKA Ladies Night
Then while tracking him down, one of her first leads tells her that Kilgrave's powers can be neutralized by surgical anesthesia.
Dr. Kurata: I've never seen such strong will. Ten hours of surgery. Kilgrave watched me the entire time.
Jessica: He was awake? No anesthesia?
Dr. Kurata: He did the whole thing with an epidural. He refused to be put under.
Jessica: He's a lot of things and a masochist isn't one of them. He didn't want to be unconscious. But he sleeps.
Dr. Kurata: It's different. Surgical anesthesia, like Propofol or Sufentanil, it shuts down different functions.
Jessica: That's it. That's his weakness.
Jessica Jones S1E2 AKA Crush Syndrome
So by the time she's in a position to capture him, it makes far more sense to use anesthesia which she knows will work. Relying on the assumption that covering up his mouth would be enough to stop his powers would have been a huge and unnecessay risk, especially since Jessica had no idea she was immune to him at that point in the story.
Once captured, the only way to prove Kilgrave's powers were real was to have him use those powers, so a brace simply wouldn't have helped, and they already had a hermetically sealed room to put him in. If they ever had proven to the world that Kilgrave's powers were real, and the system decided to incarcerate him, I assume they would have used a brace of some kind when transporting him (along with extra precautions like earplugs for prison staff). Sadly that never happened.
Finally, when Hope kills herself near the end of the season, Jessica shifts her focus to killing Kilgrave.
Kilgrave: See? I still have all the control.
Hope: Jessica, kill him now.
Jessica: We had a deal.
Kilgrave: "Abandon hope, all ye who enter here." That's the words written on the doorway to hell.
Hope: Kill him or he'll keep hurting people!
Jessica: Let me handle this.
Kilgrave: She'll never kill me. Despite her calloused, hard-bitten, and, frankly, poorly styled facade, despite her several problems, she still hopes that, at her core, she might just be a hero. But only if she can save you. The ultimate innocent victim. [...]
[Hope breaks her wine glass]
Kilgrave: You can't kill me, silly girl.
Hope: But Jessica can.
Jessica: What are you doing?
[Hope stabs herself in the neck with the broken glass]
Kilgrave: Step forward!
[The four hostages step off the edge of the bar. Jessica tears the pipe they're hanging from out of the ceiling before they die, then rushes over to Hope]
Jessica: Hope! Hope, why would you do this?
Hope: [gasping] [sobbing] You can kill him now.
Hope: Tell me. Tell me! Tell me. [breathing heavily]
Jessica: I will. I'll kill him.
Jessica Jones S1E10 AKA 1,000 Cuts
After all that happened, nobody suggested trying to catch Kilgrave alive anymore. I can't really blame them.