If it were up to me, I'd keep the cat and put the owner to sleep.

Suggestion: Can you offer to take the cat? They might be willing to give her to you. You can then take the cat to a no-kill shelter or to a rescue, or if you'd be willing, take it to a vet then find it a home. I realise the latter can be expensive, but it may be something simple that just requires anti-biotics.

Just an idea......