You need to get Joe's full respect. He needs to be able to focus on YOU, not Slick, and the only way to get his attention is by getting his respect. I, too, have a buddy sour mare, Havannah, who runs the fence when I have her girlfriend, Juno, with me, but I have Juno's respect so that she is fine with whatever I want of her, regardless of Havannah's screams. The same applies to Havannah: she is able to focus on me entirely, even with Juno calling for HER.
Since Slick is going to be a distraction, it will be hard to get Joe's attention, but it can be done. Start on the ground, with leading Joe. Put Slick in the field. Keep Joe's attention on you as you ask him to go forward, back up, turn, halt. If he's watching Slick, get his eyes back on you. Start by clucking at him to get his focus. If that does nothing, wiggle the lead rope. If he's still focused of Slick, jerk the lead. You might have to get pretty aggressive at the beginning--since he's gotten used to paying more attention to Sick than you--but stick with it. As soon as you get his attention, reward him with a stroke on his forehead or pat on the neck, tell him how good he is. If he's good about taking treats, offer him one for focusing on you. Keep working on the ground manners until you can get a turn, a halt, anything you want, and all you need is your body language and a loose loop in the rope.
Once you get Joe's attention, require it every time you work with him, be it grooming, tacking up, riding, hauling. Under saddle it may need something more to keep his attention since you're not right in front of him. In that case, ask him for maneuvers that require him to pay attention if he wants to stay on his feet: sharp turns, transitions, halts, backing up. Don't just go round and round in the same large circle, make it interesting. You want him thinking, "What's she going to ask next?"
As far as Slick goes, I guess there's nothing you can do about his meltdown since his owner seems a bit clueless. All you can do is expect the best from Joe. If you've never really required anything of Joe, it may be difficult to get the hang of paying attention to HIM all the time, too, but you must. You can't expect him to focus on you all the time if your own mind wanders. In addition, you won't be able to hold his attention unless you require it ALL THE TIME. You have to be very consistent: always ask for his focus, and reward him when he gives it to you. The rewards can become smaller as he learns his lessons, but he'll like knowing that he's doing the right thing as he's getting it.
Good luck. Once you get Joe's respect and focus, your connection with him will be so much better. You'll wonder why you never did this before. :D