Welcome! Two puppies is always hard work and you have the added hurdle of training when one of them is special needs. First I would make sure they are good and tired. When puppies have too much pent up energy they tend to use it in all the wrong ways. Take them for walks, enroll them in doggy daycare.....just something to burn that energy. Maybe get some treat toys so they are using their little minds in a constructive manner instead of terrorizing the corgi.
Does everybody have their own crate? I would suggest that so that your other dog can get away from the puppies and the puppies have a place to go when it's time for them to rest.
For the mouthing.....we have always shouted "ouch" loudly and immediately withdraw attention until they calm down.
Some training will have to be modified for the deaf pup and hopefully someone with experience on that can chime in here soon.
Good of luck and don't give up! I'm sure we have all wondered what we were thinking at some point in the puppy raising phase! Lol