How do you handle post heart surgery depression and hopelessness?

Well, as of right now, she is refusing food and meds. The rehab facility is going along with this and family has flown in to see her.

MUSIC soothes the Savage Beast...no matter what your own "beast" may be, good music has within it the qualities to calm, delight, rejuvenate (especially memory). The beauty of instruments adds to the experience. See if you cannot have a/some musicians (no matter the level of accomplishment) come to play. Nursing homes love this idea...they see it working.

From what I've learned (both husband and Mom have had open heart surgery - 10+ years ago). Both had depression to a degree afterwards. It is quite a common side effect of this particular surgery (being on heart/lung machine). Talk it over with her Dr., geriatric specialist, and even do "google" search and you will see others with same problem.

Is she on an antidepressant?

Your MIL may truly want to be allowed to die naturally, but she may also be testing your loyalty. Only you can figure out which.
The more I've gone through with my parents, my dad especially, the more I sympathize with allowing them to move to hospice/palliative care if that is what they wish. I'm also developing permanent aches and pains at age 56 and can only imagine how unhappy I might feel at age 80 or 90 if my senses fail and my body is so limited that I can't do much of anything any more.
Sometimes we think that we children should be able to provide enough happiness to make even the poorest quality of life worthwhile for our parents, but I think we assume too much. Their pride can make them feel embarrassed to be around us in their diminished condition.
Only you can tell whether the depression is a phase that can be improved, but my personal feeling has evolved to a point where I'm not sure using anti-depressant medication is the right thing to do in all cases. Sometimes, people are depressed because they are just very aware of the truth that things are going downhill and will never get better. I don't necessarily think they should be medicated out of that. I think this is the time when you honor their wishes and give them as much love and attention, and as little nagging and forcing of medication and food, as you possibly can.
I hope I don't seem like a doomsayer, but the fact is that we don't live forever. I don't think that forcing someone else's life -against their wishes- to last longer than the quality of their life and comfort is in their best interest. We all fear dying, but we also fear pain, tedium, and becoming a burden.
If the family really isn't ready to allow palliative care, you should meet with MIL's doctor and discuss her condition and prospects for improvement. As far as I know, she can move out of hospice if her health and/or attitude improve.
My mantra since the passing of my father is- when you are elderly, there are worse things than dying.
Best wishes to you and your family.

My folks had a couple of friends who went thru open heart surgery with great success, but hit a depression afterwards that mad them almost suicidal. So I know it's possible, but I don't understand what goes on in a person's body that would change them so much. I also know that before my mom died she lost her sense of taste, and she told me 'why eat when you can't enjoy it anymore?' So you have a person who is depressed after surgery, and also can't taste anymore. To me that's a recipe for disaster. She needs to be seen by someone who will prescribe her something to get her over the hump. If she has longevity in her family history, you may want to point that out to her. My mother-in-law is 88 years old, but she could very well have another 10 years left because her dad (who she takes after) lived to be 100.