You can read through the information from the Bazelton Center (cbg provided this earlier but here is a clickable link):
https://www.in.gov/ihcda/files/Emotional_Support_Animal.pdf
If you wish to fight to extend your month-to-month tenancy or to sue your landlord for discrimination (which to me seems a futile and potentially costly task), you must first learn whether your landlord owns more than 3 single family rental houses or not (if not, FHA doesn’t apply).
Additionally, you must be prepared to prove your disability (its existence, not necessarily the specific disability) and to demonstrate that the emotional support animal is necessary for your use and enjoyment of the rental.
You should be prepared to give up all of your cats and (probably) one of your dogs and pay for any damages these animals have caused (and possibly pay any fines for violation of the no-pet policy). The 2014 lease terms are the terms under which your month-to-month rental is governed.
I hope your emotional support dog (singular) is good-natured and not a nuisance (vicious, destructive) because you can be held liable for dog bites or damages to the house, and these alone can be justifiable legal reasons for early termination of a lease.
All communications with your landlord should be in writing. See the information in the link above for what needs to be done.
I believe your current landlord has good support for termination of your month-to-month rental and these reasons are justifiable even if you have the medical need for an emotional support dog.