How long does the smell of urine linger? Does it disappear naturally?

I refer to a book I read a very many long decades ago. Hints of Heloise. Search for that first.
My own common knowledge says to have the carpet steam cleaned, by a professional carpet cleaning company. Please stop putting more liquid into the carpet problem. Use a dry compound like baking soda, and leave that on the carpet. Remove within four hours.
The smell could be gone or has seeped deep into the carpet underlay fabric or foam. Only a deep cleaning with a professional system will get down and pull the excess offending aroma from there.
Once again, Hints from Heloise. Worth the read for all of the home problems.