Was the questioner wrong for not telling anyone about her sister's illness after she was asked to do so, and what should she do to correct this mistake if she was wrong?

Treatment is permissible and recommended, and it may become obligatory if its ترك leads to destruction, based on the Almighty's saying: "And do not kill yourselves. Indeed, Allah is ever Merciful to you," and His saying: "And do not throw [yourselves] with your [own] hands into destruction." So, if the illness leads to death, then silence and not informing her family is considered aiding her in abandoning an obligation and surrendering her to destruction. Concealing the matter of her sister's illness until it spread and overcame her is considered abandoning an obligation and committing a sin, and the atonement for that is repentance to Allah, the Almighty.