As mentioned, consulting with a behaviorist is a good idea. Before allowing anyone to work with your dogs, watch them work with other dogs to see if their style is a good fit for your family.
I would also make sure both are checked thoroughly by a vet, bloodwork (including thyroid testing) included, to make sure that health issues aren't making one or both uncomfortable. I know of times when a basenji sensed a terminal condition before it even showed in bloodwork. Are they both spayed? If not, going into season may be affecting either one's behavior.
While Mia heals and you look for the right behaviorist, keep them separated. This may require two layers of fencing to make sure that they don't start fighting through the fence. I've had over 200 basenji fosters and have kept newbies behind a layer or two of fencing while they learn how to behave here. Particularly when the more challenging ones arrive, nothing is for free, including access to human furniture. They get 'their spot' in the house (in their pen where they can still see the others) where they can feel safe and not worry about others bothering them. I build up communication and confidence by starting (or going back to) basic commands with every meal (with food sitting at their feet): sit, stay, leave it (a VERY important command), look at me, eat. Go for walks together, each human holding one, keeping them separate, yet close enough to be aware of the other. The outside world can be a good distraction to keep them from focusing on hating the other. When I introduce (or reintroduce) them to other dogs in a securely fenced yard, I start at times that aren't so exciting - quiet times, nap times, etc., not around feeding time or when we have visitors. When they are relaxed, you can work on trying them in the house with both of you there, each holding one on a leash. It is important to not make a big deal about it - just sit in a room together, using the leash to remove one to her safe area if she shows any apprehension or aggression. Go very slow. Don't rush it. That is when failures happen.
I hope Mia heals quickly.