The whole, "you always loved her best" really makes me throw up a little in my mouth when it comes from adults who are also over 30......
If you must send a bday card, do it but keep it short and to the point of her and her birthday. Maybe write a quick blurb about your willingness for the two of you to sort out current differences but that you cannot go back in time and change anything. Always focus on, the "where do we go from here" concept. Acknowledge her pain or disappointment and then let it go where it belongs, under the bridge along with a sea of other past moments.....
Serious illnesses are so hard on everyone and even tho the patient has it so much worse. YD may be seeing it as her own hereditary prophecy. I lost a few friends when i got so gravely ill and lost my feet. They just couldn't see me that way because it reminded them of their own vulnerability. Losing body parts is an awful thing to face for all.....
Hugs to you dear sister in struggle! My hopes are with you and your family for a speedy and complete recovery.