The backstory:
I'm 32, DSis is 25 and our mum died 2.5 months ago after a brief illness (metastatic breast cancer which we thought had gone away. It hadn't).
I live in the South of England with DH and our two DC (3yo and 2yo) and both myself and DH work FT. DSis is currently taking a break from her uni course (made this decision when she realised DM was ill and will be returning in September) and is working part-time in a deli in the small town where my mum lived. DFather is not from UK originally and was here for DM's treatment but has since moved back to his home region (not his actual country but we lived there as a family for a long time). So I am 4 hours away from DSis and DF is even further than that. She has a good circle of friends, she tells us (haven't met any save one or two) but isn't dating anyone that I'm aware of and has no DC. We have other relatives in the small town (our aunt and grandfather).
The problem I'm having is related to the fact that DM let DSis get away with what I considered to be quite rude behaviour - avoiding relatives, acting like the world revolved around her, etc. DM was DSis's major defender whilst the rest of us got progressively more and more pissed off at her childishness and selfish attitude (i.e. not participating in the once-in-a-blue-moon social visits, sitting there sullenly and not engaging when she did turn up, suddenly being sweetness and light when she wanted company from you but leaving as soon as her friends made her a better offer, etc). We didn't demand affection on tap, but an indication of occasional willingness would have gone a long way.
As a result, I feel that DSis is probably the most affected of all of us by DM's death. I feel like I want to try and look after her (because she is my little sister and I do love her) and so I do occasionally call/text/FB message to check in. She's generally less than enthusiastic when I get in touch and will reluctantly respond to my attempts at conversation, but she complained to DF that he didn't call her enough the other day 
and he was a bit bewildered because he says that she never seems to want to speak to him either. The rest of the family would like to be friends with her but are somewhat tired of her attitude and have more or less given up trying to make contact.
I worry about her, because in some ways she is very very hard on herself and criticises her own behaviour relentlessly, but in other ways she completely explodes at even a hint of implied criticism, as in:
Me: "Oh, I went to see DAunt today. She asked how you were doing."
DSis:"Oh yes, I KNOW, I haven't seen her in a while"
I don't want to have conversations with her that she'll use to beat herself up with afterwards, but I don't want to constantly be swallowing my bile at her not-so-passive-aggressive arsiness either. She treated DM like that and DM's advice to everyone else was to 'just ignore it because she's so lovely when she's happy'.
Sigh. I just don't know whether to persist in trying to talk to her or to just leave her the fuck alone.
Any advice would be appreciated.