I am in a similar position to your sister. My sister is the golden child of the family, and my parents just can't do enough for her. I get almost no practical or emotional support at all. In my case, the feelings of hurt that this generates are exacerbated by the fact that my sister does exploit my parents a bit (lives at home despite having a job that pays ££ because she feels she 'deserves' more than a starter home - DP wait on her hand and foot). Additionally, my grandmother is very needy and quite exhausting to be around - so between those two relationships, my parents simply don't have any energy or emotional resources left for anyone else.
Also, perhaps because of age, my mother has started to find other people's emotions difficult to deal with and to diminish them. So regarding the death of my grandmother's partner a couple of years ago, she will say that they were 'only' together for two decades in their old age, so it isn't much of a loss. If I try to raise my inability to have children as a subject, I'm told that it's a good thing because I wouldn't be a brilliant mother and children are too much work for me etc. etc. etc.
I have had to learn to be completely independent and do absolutely everything for myself, and it has not been easy at times. If I am completely honest, the fact that I am now reasonable settled financially has also played a role in my being OK with it. When I was grafting and having to live in horrible places while my sister was comfortable at home, the contrast was quite start and ever-present. I love my family to bits and am now accepting of the situation, but it has taken me until my mid 30s to get here.
Obviously, this affects the advice I'm going to give you, which is: you cannot underestimate the impact that this has had on your sister, long term. I'm not for a second saying that it is your fault. You are clearly a nice person who is really trying to rectify decisions that were not of your making - but you are still a constant reminder to her of how she wasn't loved. The fact that you are settled in life and she is not may have a lot more to do with that than you realise - and will also be something she's constantly dealing with.
The fact that she's written this letter is a howl against her unfair universe. Of course she shouldn't have done it - but then again, she shouldn't be in the position she's in either. It seems to me that she's very deliberately taken a course of action that forces this out into the open. If you do want a relationship with her, you probably need to recognise her universe, not to ignore it or pretend that this is an equal relationship. One of the most powerful things you can do is to affirm that it is wrong and unfair that she's been dealt this hand. The relationship can only really go forward if you build up from that foundation, rather than pretending that you can somehow neutralise it.
In my 20s when I was hungry, poor and depressed I actually sat down and drafted a letter about my situation, with the vague idea of sending it to the Guardian. I didn't, thank goodness!! But I think I can understand why your sister wanted to do it. When you are the scapegoat who is always, always outside of everything and never good enough, it is like you disappear. Your position, your point of view, your very being in the world feel tenuous. It is like you could just dwindle away and no-one would really notice because you are never fully 'present' in the world when you are in it - you never 'count' for as much as other people. Not being able to have children could powerfully accentuate those feelings, because it is in many ways a very powerful and creative affirmation of self to bring a new being into the world and bring them up - and, once again, it seems that it is sadly shut to your sister. I am not surprised that your sister is suicidal, to be honest, because it is kind of the logical conclusion of this way of thinking that has been produced by very bad parenting. She must feel like not only your parents, but the universe thinks that she's not good enough. I would take her depressed behaviour in this regard extremely seriously indeed. I don't think self-harm in her case will be an empty gesture.
On the upside, this actually means that you have a powerful weapon in your arsenal, which is to recognise the legitimacy of your sister's position. When I think back, what I needed - desperately - was for someone to turn round and say to me 'You know, this really, really isn't fair'. Don't underestimate the potency of that, and be careful with it. When I finally decided to get therapy, just having someone recognise my situation was really powerful and cathartic - I actually went completely hysterical when my therapist said it. It was like being struck with emotional lightning, to the point that I couldn't breathe.
I realise that you will also have been the sufferer as the 'golden child', and that this is not the enviable place to be that it might seem from the outside. But I think there is a difference perhaps in the degree of harm that has been caused to each of you that is very significant and important, though painful, for you to recognise. I wouldn't say this to many posters, because there are a lot of people who are entirely self-absorbed and self-righteous in their victimhood, but you seem genuinely caring, to the point that you would be willing to accept an emotional burden yourself if you needed to. Think about it.