You speak of a creature that has come to trust you, and now his strength fails. This very picture mirrors our own estate apart from God. All strength apart from God is derived strength and is, consequently, measurable. Yes, apart from God it is not strength at all and, consequently, must come to an end. Some signs of age, the creatures show already, but the time will come when their strength shall utterly fail. You have done well to seek the easing of his suffering, for we very speedily care for bodily diseases; they are too painful to let us slumber in silence.
Yet let this affliction turn your hearts heavenward. The leaves of the tree are for the healing of the nations, and there is an abundance of healing power in Jesus Christ. If you feel for a poor suffering dog, how much more does God pity the sin-sick soul? I draw your attention to this for your comfort, there is enough in Christ for every sin-sick sinner. When all your own strength fails you, your strength shall be to sit still while God arises and becomes your arm every morning. To those who have no might, He increases strength.
You say it is amazing how an animal becomes part of the family. Have you not felt, when one whom you very much esteem is sick and ill, that you would be glad enough to bear the pain? So cast this care, and every care, upon Him. The glory of our strength must never lie in anything, it must lie in God alone. O you who are now sick unto death, ask to be forgiven and healing will come from the pardoning hand. I will strengthen them in the Lord. Then, indeed, the inhabitant shall not say, I am sick.
May the Lord mercifully sustain the creature through the night, and give you and your wife to lean entirely on divine strength. God bless you.