I'd wait until friend gets back, rather than risk making it worse. At least if they take action and make it worse, they're responsible. You were doing them a favour and their cat threw-up. You attempted to clean up, and it didn't work out. Same thing could have happened to them. It's not like you had a party in the house and are trying to hide it.
It might help to distract them with something else, e.g. the sick cat and how looking after it was your number one priority! Alternatively, get the cat to throw up on the rest of the carpet, apply the same "cleaner", and achieve a uniform new colour.