Prayers for comfort and healing up for you and your wife.
Don't second guess yourself about what you 'could' have done - if you'd done something else you'd be second guessing that too because you're hurting right now. But it is better for Copper to have passed with you beside her, in her own home and surrounded by love, than in the vet's office. They do love the animals they care for, but it's not the same as being at home with family. And she would have been uncomfortable, and it sounds as though her heart just gave out and they wouldn't have been able to do a thing for her.
Don't worry about grieving, it's to be expected and it won't always be so heart-wrenching. You'll remember the good times and be able to remember them happily. It just takes time.