You can't help people who are more invested in their pain and misery than they are in life and their family. She has a self-imposed loneliness - her friends can't stand her, and apparently neither can your father. She says she wants to be left alone, but she doesn't really want that or she wouldn't be forcing a 4 week visit and constant phone calls.
She might need medication - we can't judge that because we haven't met her and we aren't doctors. There are some amazing things with new research in battling inflammation and immune suppression - there is a new product she can use for fibromyalgia that is manufactured under strict conditions and of high quality. I'm in this field and can help you with more info. A lot of people get results the next day, while others take a few weeks. But that's not going to solve all of her depression, although if she's pain-free she might be able to enjoy things more.
You definitely can't sit around for 4 weeks, and your kids should not be subjected to that. If you and they are her reasons for living in HER mind, that doesn't mean you need to accept that burden. Do not let her guilt-trip you into passing on more misery to yourself and your children. Your children will resent her, and you, for the rest of their lives. Do not do to them what your mother has done to you.
If they won't pay for your tickets unless you stay for 4 weeks, just don't go. You can tell your parents that the environment is too toxic for all of you, and that if they insist on living that way, you can't be part of it. I would consider going more often but for very short periods of time - depending on how far away your parents are.
She's getting a pay-off for her behavior so it's in her interests to continue it. And you are enabling her by being her only sounding board. Because she doesn't speak much English and the kids don't speak much Spanish, you have put yourself in the middle as her only relationship.
I think you can tell her that it's too much for you to visit such a sick person for 4 weeks and inflict more pain and stress on her. And tell her it's no good for anyone to be miserable for so long a period of time - not her, not your father, not you, not your kids. If you do go, you've got to get to the point where you walk out of the room when she complains. Give her 30 minutes, 15 if you can, and then say it's too stressful for her and you are giving her a break so she and her pain can rest. Leave every time. There is no purpose to all the talking - it cures nothing. She is creating and maintaining the divisions in the family - you don't have to participate.
I realize that there are cultural considerations but perhaps you can consult with elder services (depending on her age) or a local psychologist or social worker to see what therapies are available and what services could be provided in her own home (personal care people, cleaning help, etc.). Let her know that, if she's that sick and lonely, she needs help.