You need to take the patient to a grooming table and in the clothes option (I think its a bling collar icon) you apply bandages like babybop said. I apply a bandage daily but I am not sure if you keep the same bandage on or not.
And yes it does get boring after about a week. I moved to larger premises which made the game almost impossible because you cant seem to get around all the pets on the same day and you get penalised and cant treat them in the alloted time. It is only broken bones which you need to keep in for 5 days, the rest you keep in for 3 days unless you think you can treat an ugly or smelly animal on the same day it comes in.
The neck cone can only be applied on the grooming table and not the examination table.