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We got him started this evening on prednisone. Will see how much time that buys him.
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My experience has been a month, maybe two. Keep in mind one thing: After the prednisone stops working, and they start to crash, they can crash hard and fast. So, here is my unpleasant advice: Be prepared to have your pup put to sleep either BEFORE they crash, or be prepared to do it very quickly once they start to crash.
It sucks, and sucks hard. I helped my debilitated mother through a very similar situation with her dog, whom I also loved very much. IIRC, we planned on one month, and had the euthanasia date set already. It just so happened that he started to crash just the day before, so our timing was not terribly off. Having them put to sleep while they still feel good can be even more emotionally taxing, but IMHO, it's a usually a more selfless thing to do than keep them around to suffer.
My condolences and sympathy, it's a tough situation to be in, no matter how it goes.