Feel guilty for my mothers death

Sorry for how long this is i just wanted to rant

My mum just died of lung cancer that spread to the liver just over a week ago. I feel so shit about it. I was separated from my mother for 10 years(ive just turned 19 now) and was only allowed to see her once a month. And all that time I resented her a bit bc she would only phone me whenever my sister was coming round (she was also separated from me) everything was always about sister even while I used to live with my mother. I barely ate, she barely talked to me, she didn't really seem to care much. I was always neglected bc of my sister's needs.(she has mild learning difficulties) my mother was a heavy smoker and we always ran out of money so i woufl always save pennies and count them so my mum could buy some food...or well tobacco. I would try help my mum bc she was sick herself mentally and physically and help my sister all the time as a kid so i didnt go school much. I was ill a few times myself but hey im alive. So yh I was taken away when I was 9. And it wasn't till I was 18 I was able to see my mum whenever i wanted so i would on weekends. And I finally got to see my mum's love for me a bit more but I still wasn't close with my mum. She didn't talk much, she never did. She had her own mental health issue and was genuinely also ill all her life. But I found out she did talk with my sister, like actual responses and not just short answers. And I hated going to her place. It always brought bad memories of when she tried to overdose and smell made me sick. I always cleaned the place whenever i came. She was always smoking nonstop. She had carers that were meant to look after her and make sure she was taking her medication and clean, but they didn't really do her job, but then again, my mother was really stubborn. She barely went to doxtors appointments or told anyone if she was unwell or in pain. Either way, my sister (she's 24 now, moved back in when she was 19/20 she was only separatedfor a few years, they had taken away from my mother a few years after i was.), has moved back with my mum for a few years now and this year I received a message my mother wasnt eating meals and just eating snacks. I brushed it off and just told my sister to just make her food to eat. Bc my mother was always like this, just eating crap and not food, I then said to book a doctors appointment after I called mum, and it sounded like she had a bad cold. Her breathing was off, a bit chesty, just like whenever she had a bad cold. A week passed, and it was finally time for the doctors appointment, which her carer was going to take her too, but I didn't bc my mother didn't let them into the house. So that weekend I went to the house and was shocked at what I saw. My mother couldn't walk, had lost so much weight since I saw her last, which must have been like 3 weeks. It was horrible. She was in such a state I could go on. And couldnt believe all things my sister left out to tell me. While this was happening, my sister was busy going out with her friends, so I called 111, and they sent a doctor out at 4am, who then called an ambulance. He believed she had a chest infection, which made sense. My sister then went out again that same day while I went with my mother to the hospital and stayed with her. She was then taken to ICU in the evening, and the next day, they told me she had cancer and would only have a few days or weeks. I managed to get her into a hospice for her last few days as I watched her die. She was very confused at times and pretty much only remembered my and my sisters' names. Her family came to visit too which was nice.My sister stayed at the hospice with me till her death. I knew my aunts(her sisters) blamed me for her death, and I do feel responsible for it. I should have known something was seriously wrong. Should of done something sooner. Either way, im sorting out the funeral(which is the hardest bc I dont even know what to say) and everything that is left, helping my sister get a new home bc she wants to move. my sister stayed with me so she isnt alone. Working and going uni as much as I can bc I still want to keep some normalcy even if I feel guilty for wanting to continue to live my normal life