She is sick and does not understand anything and is not aware of her surroundings. Does she have to feed the poor (in lieu of fasting)?

In order for fasting to be obligatory, it is stipulated that a person be of sound mind, because the Prophet (peace and blessings of Allaah be upon him) said: “The Pen has been lifted from three: from the insane person until he comes back to his senses, from the sleeper until he wakes up and from the minor until he reaches puberty.” Narrated by Abu Dawood (4399); classed as saheeh by al-Albaani in Saheeh Abi Dawood. 

As your sister has reached this stage or being unaware and unable to distinguish anything, she is not obliged to fast, and she does not have to make up missed fasts or feed the poor instead. 

Prayer is like fasting; she is not obliged to pray either. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said in al-Sharh al-Mumti’ (6/202): 

Anyone who is not of sound mind is not accountable, and he does not have to do any of the duties enjoined by Islam, such as praying, fasting, feeding the poor etc, i.e., he does not have to do anything at all. Based on this the one who is feeble-minded does not have to fast or feed the poor, because he has lost the thing that qualified him to do that, namely his reason. End quote.