It sounds like its a choice of keeping the dog AND the mice or getting rid of both. my sister-in-law has a Jack Russel and a cat. They get along. I suggest that if training doesn't help, you find a new home for the dog.
1. Traps are very effective but you will need to check them several times a day. The cheap spring traps work very wwell. Tie bacon on with fish line. Place the traps where you think the mice are getting into the walls. Do you have a basement? The top ledge of the foundation is a good spot.
2. A cat isn't enough. You need a mouser. Not all cats make good mousers. Siamese are the worst. Did you know a cat has to be trained to be a mouser? Usually momma cat does this, but the momma cat had to be a mouser and had to have time to train the kitten. You can train the cat to be a mouser.
3. The sonic devices are only good at removing $$ from your wallet. If they were really effective - your dog wouldn't be happy either.