Where I used to work, there was a dead rat somewhere in the walls of one room. The smell was just awful, and it lasted for weeks, if not months. Short of tearing the walls apart one by one until the rat was located, there was nothing that could be done.
I have a dog who can detect termites. We were amused, until we saw the bill from the termite man! Maybe we should offer to rent him the dog!
The smell of dead rat is the price of poison vs traps.
I just had to comment to your post. Somehow almost every year a mouse will get between the walls of our house and die. You are so right about the smell. It is just awful. It is this sickly sweet smell and it is very unique - nothing else smells quite like it. It always happens during the summer and the heat will make it so intense that I have literally gone crazy. My husband has done his best but there is no way he could ever find it.
I just chalk it up to living in the country. I never had this happen when I was a “city” girl.