A California landlord has the duty to maintain the rental in a habitable manner. Under the "implied warranty of habitability," the landlord is legally required to repair conditions that seriously affect either the health or safety of the rental unit's occupants, or fails to substantially comply with one or more of the state and local building and health codes. "Habitable" also means the rental unit meets the standards set forth in California Civil Code Section 1941.1.
You should immediately: (1) take pictures; (2) call the housing department and the health department in your local city; and simultaneously (3) WRITE a letter to the landlord explaining that under California law he has a duty to maintain the premises in a habitable condition. California Civil Code considers rodents and their feces as creating an uninhabitable situation. Dead decaying rats and their feces not only poses a significant risk to the health and safety of the occupants, but is also leaving a tremendous stench. Add a paragraph stating that he has been on notice of the situation and his prior efforts are unacceptable. End the letter with a statement that you expect that he will rectify the problem within 72 hours by using a reputable pest control service to remove any living and all of the decaying rats and their feces, and that if he does not you will consider your legal options.
I would strongly suggest you contact a local landlord tenant attorney in your area, as you have various options each of which requires you to do various things and which would entail to lengthy of a discussion in this forum. Some of your options include: (1) paying less rent or withholding the entire rent until the problem is fixed (place rent into an escrow account); (2) hiring someone to make the necessary repairs, and deducting the cost from the rent; (3) moving out, even in the middle of a lease, and suing for constructive eviction; (4) suing in Small Claims Court; and/or (5) calling the local building inspector, who can usually order the landlord to make repairs. 1 to 4 can cause a landlord to sue you for eviction, so please consult with a local landlord tenant attorney before proceeding.
Again, you should NOT attempt 1 - 4 above until you have retained a local landlord tenant attorney who can provide specific advice based on the laws of your local and prepare the necessary notices and letters.
Disclaimer. The above is provided for informational purposes only, does not constitute legal advice, should not be relied upon, and does not create an attorney client relationship.