You want to make contact with your sister as you still care about her. Ignore those awful comments about you doing so for your own benefit.
Who told you she was ill?
I like the idea of sending flowers or a plant with a note saying sorry she is ill and offering help but maybe flowers are not you and a letter seems more natural. Problem with a letter too many words that can be misinterpreted. Read and re read, discussed and ignored.
I also think her family will appreciate and be comforted by your trying to make contact.
After all she is your sister and you do have shared memories that others even her husband and children don't have.
For the people who had such awful sisters, I can understand your position. Never spoke to you, today people would say she needs counselling, then a clip round the ear.
Another Tory MP crosses the floor
NHS U turn on trans terminology