All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
First of all, we ask Allah, The Almighty, and the All-Capable to cure your wife from this illness, as He is the Healer (the Curer) and there is no cure except through Him. He is capable of all things, and every matter is easy for him.
Hence, we advise you to supplicate Him earnestly as Allah Says (what means): {Is He [not best] who responds to the desperate one when he calls upon Him and removes evil and makes you inheritors of the earth? Is there a deity with Allah? Little do you remember.} [Quran 27:62]
Your wife is obliged to obey you about what you ordered her to do, because this is part of her obedience to you in what is reasonable [which is about getting rid of the cat].
Abu Hurayrah narrated that the Prophet said: “If I were to order anyone to prostrate to someone else other than Allah, I would have ordered the wife to prostrate to her husband.” [At-Tirmithi]
This is the Islamic ruling, but we hope that you will take into account the health condition of your wife and avoid as much as possible what saddens her. You may reach a compromise.
Instead of getting rid of the cat, you may keep it in a cage so that it will not destroy and mess things around in the house; while giving it food and drink, as a way of avoiding the threat that is mentioned in the Hadeeth.
Abdullah ibn ‘Umar narrated that the Prophet said:
“A woman was punished and entered Hellfire because she kept a cat in captivity until it died; she neither fed it nor gave it water to drink; nor did she set it free so that it could eat from the vermin of the earth.” [Al-Bukhari and Muslim]
Allah knows best.