Awwwww, how rewarding!!! I have 4 dogs and one rescue kitty, and do dog rescue as well.
As you probably know, no self-respecting cat will vomit on a tile floor in the middle of the afternoon. Better to do it in a secluded part of the house at 3 am - and definitely on the carpet.
If you don't already use it, try Nature's Miracle or Simple Solution. (I use Nature's Miracle) It will save you a boatload of time and effort. It is all-natural and enzymatic and works WAY better than baking soda or any commercial carpet cleaner I have found. It cleans up EVERY kind of pet and child accident you can think of. (Just ask how great it is at getting blood out of the carpet when your DH decides that trimming dog nails in the living room is a swell idea!

) I know I sound like a commercial, but the minute I put it on the blood and just blotted it out, I was sold. I even use it to clean my tile floors.
Again, kudos to you, kitty-rescuer!!!
Tracy