Mix some healing oil, using:
6 drops Lavender oil
6 drops Camphor oil
6 drops Rosemary oil
1/2 ounce base oil (apricot kernel, jojoba, grapeseed, even olive oil)
Annoint a black candle, a red candle and a brown candle with this healing oil.
The black is for absorption of the negativity (the illness), the red is for strength and health, and the brown is an "astral" or "significator" candle for the animal.
Place the brown candle between the black and red candles, and make a ring of stones around the set-up, using stones or crystals that you associate with healing (amethyst, quartz, agate...you choose whichever type or combination of types).
Place some of the oil in your hands, and begin to rub them together, generating heat and energy. When you feel that you've built a small "bundle" of healing energy, place your hands on your pet, stroking it and giving it the healing energy you've built, making sure to concentrate on any areas where you feel or sense the illness.
Focus on the candles, continuing to maintain contact with your pet, and visualize the strength from the red candle pouring into the brown candle, and pushing all the sickness into the black candle. Attune with the Goddess, and when you've got a good grip on the visualizations above (and the animal!), say:
"Goddess, with your healing touch
Bless this animal we love so much.
God, with beasts as your domain,
Remove the sickness, heal the pain.
So mote it be!"
Continue the stroking and energy flow until you feel that the spell is done, then allow the animal plenty of time to rest and heal.
Obviously, if you wish to perform this spell in a cast circle, that's your choice.
If you wish to continue the spell for a number of days (I'd suggest at least 3 days), and your pet has a special place where it rests or "lives" (such as a cage, etc.), ring this area, if possible, with the stones, place a red candle on one side of the area and a black candle on the other (annointed with the healing oil); focus on the red candle and project its energy into area with your pet, visualizing the illness being pushed out the other side, since there's no room for it, and into the black candle.
Rub the oil onto your hands and stroke it into/onto the animal's body as often as you feel necessary.