Praise be to Allah.
Down’s syndrome is a genetic disorder; it includes delays in the development of cognitive ability and physical growth. The external signs of Down’s syndrome include a crease in the eyelids, short neck and hands, and poor muscle tone.
For fasting to be obligatory, it is stipulated that one should be of sound mind, because of the words of the Prophet (blessings and peace of Allah be upon him): “… The Pen has been lifted from three: from the insane person whose mind is deranged until he comes to his senses, from the sleeper until he awakens and from the child until he reaches maturity…” Narrated by Abu Dawood, 4399; classed as saheeh by al-Albaani in Saheeh Sunan Abi Dawood.
If your sister is affected by a delay in cognitive ability to the extent that she has not been able to reach the level of discernment, and she does not understand the teachings of Islam, then in this case she is not obliged to fast or to make up missed fasts, and you do not have to pay the fidyah on her behalf, because she is not accountable in the first place.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in ash-Sharh al-Mumti‘ (6/324): Anyone who does not have cognitive ability is not accountable, and none of the obligatory duties of Islam are binding upon him, such as prayer, fasting and feeding the poor instead of fasting. In other words, he does not have to do anything at all, with the exception of financial obligations. End quote.
And Allah knows best.