Phone her.
Say to her calmly and without raising your voice over the phone why she was under the impression that when she was allowed to stay in your house while you were on holidays, she thought her dog was too? If it was under the weather, she should have stayed at home and looked after the dog, not bring a sick animal to your house. Also, if it was sick enough to have diarrhea, it should have been left in the kitchen with newspapers all over the floor so it was easy to clean up after. Not in a carpeted bedroom.
Ask her what she plans to do to sort out the issue? Her nephew is sleeping in a room that her dog defecated in and the stench is unbearable, especially a few days after this happened.
Let her answer.
If her answer doesn't match what you would expect from her (either paying to get it cleaned, paying the hire costs of a cleaning machine or replacing the carpet) then you have to set out your stall and say that you're not happy with that and tell her what you expect from her.
Then, contact your parents, tell them what you've said to your sister and say that they are not to pay for whatever your sister has agreed to as it is her mess, not theirs to sort out and your just letting them know in case she approaches them to pay for it.
Next time, don't let her stay or if you do, lock the rooms that you don't want her in and set clear house rules in relation to pets and visitors.
Your poor son. That is utterly disgusting and unacceptable.