I would never throw a cutting away until it's less than an inch long
Cut until you see white wood. You have to make sure all the rot is gone. It's best to sterilize your knife or clippers between cuts and yes, giving it the peroxide treatment is a good idea. Then dip in rooting hormone. Then let the cutting callous for a week before planting again. A very light perlite mixture is good.
If these are very small or very precious you might consider grafting them, too.
Good luck! Keep us posted.
