I went through a similar situation when I was a kid. We had a black lab mix named Samantha who had a thyroid issue that resulted in her being about 50 pounds overweight. Yes, a 130 lb. black lab if you can imagine such a thing. Over the years, the added weight took it's toll on her and she developed hip dysplasia. The effects of the condition eventually getting to the point where she struggled mightily to do common tasks like picking herself up and walking. When she looked at us we could tell she was miserable and in pain...
I still remember the day my parents made "the call" to the vet. How upset I was and my mom explaining why it had to be done. The vet knew our family well and made the trip to the house. We took her outside to the patio and had her lay on her favorite blanket. It was a beautiful day. I held her, my mom petted her, and she went to sleep for the last time.
A year or so prior to this, my parents had taken in a one year old golden retriever. Having him around definitely helped our family get over the loss of Samantha.
So I guess what I'm saying is that you and your family will get through this. It won't be easy, but you will. And, like others have said, another dog will go a long ways in the healing process and help you all move on. When you're ready of course. After all, who doesn't love a cute little puppy?
It's like George Carlin once said... "Life is a series of dogs." Like most things he said, he was right on the money.