If someone is sufficiently ill on Yom Kippur to be halachically allowed to eat, would there be any value in him fasting for a whole day after he gets well? I'm sure it would be voluntary, but is it ever encouraged? Anything from the Sources?
No, beacause yom kippur is a din in the day - we fast on the specific day of yom kippur. While fasts like tisha ba'av or other derabanans can be pushed off (called a nicdche), yom kippur is only on the specific day. That being said, there is a story with a famous gadol (forget which), that when the doctor told him he wouldn't be able to fast on yom kippur, he said something along the lines of all these years I did what Hashem wanted by not eating, but this year I am going to do what Hashem wants by eating. Just food for thought.
Answer one is absolutely correct furthermore we only find this concept of making up a fast in a fast which we took upon ourselves. That which the Torah or the rabbi's command are for a specific date and if one can't fast on that date then there is no fast. Making up a fast is only when we promised to fast on a day not obligated by the Torah or rabbis and that obligation comes from our own neder or vow. Our own vows we can make up, not specific perscribed days. This is because our vow doesn't really mean specifically that day, but to afflict ourselves as an atonement and for retrospection. This can really be done any day therefore can be made up. Vows can be made up time doesn't return.
In general, if one cannot fast Yom Kippur and is directed not to fast by his/her LOR then it is likely to be considered a case of Pikuach Nefesh and it is even considered a mitzva to break one's fast to preserve their life!
I dont have any sources off hand, but it is common knowledge in mainstream Judaism that when once is patur (exempt) from something such as fasting, they do not need to "make it up" as their obligation to fast has been discharged.