It sounds like she's afraid of the label. She obviously realized that something is wrong if she's calling you crying all the time. Maybe if you could help her see that her situation could improve without calling it depression.
The book "The Natural Guide to Pregnancy and Postpartum Health"
taught me that almost anyone going through pregnancy and nursing (especially several times in few years) will have some nutritional deficits and some decrease in mental health. This author's answer is a specialized diet when I would rather use supplements (I just think it is too hard to get enough of the nutrients through food). But if you could convince your sil that you don't think she's crazy, you just think she's taken a hit from her situation. And she would likely feel better with, for example, better sleep, exercise, outdoor light, more personal time alone and some supplements, would she do it?
Since it sounds like her dh is on board, he could help a lot in making that happen. I would also send him to postpartum dads
If you really want to push for the doctor, I would suggest asking her to take a test like the one in our sticky and finding some testimonials on the web (you could find some in this forum, I bet) from moms about what their experience with ppd was like and how they were much more like themselves again after getting treatment.
But in the end, you can't force someone to get help. Good luck. I've been in that situation before, trying to convince someone that there are ways to treat ppd, you don't have to live like this. It is really hard.