Grief at the loss of a loved one is one of the most powerful emotional processes we can experience as human beings. Well studied and described you can look up the psychological descriptions of its different stages. However, intellectually understanding it has little impact on the person in the throes of of desperate helplessness of the shock of losing someone you care for.
Pets, for the most part, live a compressed lifetime in comparison to us. The time line of their maturation and natural aging occurs in a matter of a decade or two and for us the memories of them being young and lively is very fresh in our minds and recent in time.
With our companion animals, we maintain a caregiver role their whole life long, there is no real "growing up", we see to their every continuing need. How then to reconcile that we could not make old age, disease, or decline all better again?
Honor their life and importance to you by gathering together the accoutrements of their presence and memories - photographs, a favorite toy, video clips burned on a disk, collar or tag, etc. and put them in a memory box to place prominently in your home. If you need to visit these things they are there to look through, otherwise they serve to represent the energy of the the entity's life and presence in your heart as well as home. Unfortunately we are the ones that are left behind.
When our cat also had come to that point and the same decision had to be made, we brought her back home stopping at a department store to select a decorative cardboard box for interment. I also bought a smaller one of the same design/pattern and it is still here in the house with a few mementos inside. I am glad to see it when I run across it and think very well of her at the time (along with a bit of melancholy).
It does get better with time. The pangs of loss remain but become manageable - they are the lasting measure of the importance that individual creature played in your lives.
---------
There is, of course, another story here - a hero's tale!
One fateful day your husband reached down through trash and detritus and plucked that being out of abandonment and certain imminent death. They lived a long and loved life with the trusted companionship of a loving family.
Do you not think that being was not ever so grateful for her own rescue by him? She showed it daily. When the time came to relieve her of experiencing the severe pain and panic of her body failing, he again made a difficult and important decision. With him close by she passed a dignified and much less stressful death. Losing consciousness and falling asleep in trusted company.
Give him lots of hugs and spend time talking it out, tears are good too.
It does get better.