I am presently being hospitalized in an eating disorder unit. There are very minimal kosher options. However, they do allow bringing in food. This is very complicated and annoying since my family hardly comes, and we are not near any Jewish bikur cholim, etc. What level of kashrus am I mechuyav in keeping, and what can I eat from the hospital? What is the halachah concerning foods that have kosher ingredients, but made in a non kosher kitchen?
As a choleh she’ein bo sakanah (a sick person whose life is not endangered), you would not be obligated to follow different chumros in kashrus, but you may not eat food that is not kosher. If food with kosher ingredients is cooked by non-Jews, some permit its consumption for a sick person, but only where there is no Jew available to cook food. Others are stringent even in this matter. It is therefore preferable to avoid this food, but if it is difficult to do so, and there is nobody to cook the food (or even to turn on the fire), you may eat it.
Sources: See Mishnah Berurah 328:63. There is something of a contradiction in the rulings of Mishnah Berurah on this question (see also 318:14), and some distinguish between a person whose life is endangered and one who is merely sick (see Zichron Yehuda 1:13).
The source of the lenient opinion is Rema, Yoreh De’ah 113:17 (based on a dispute between Rashba and Ra’ah; the discussion is over whether a food cooked by a non-Jew on Shabbos for somebody sick is permitted even after Shabbos), and see Nekudas Hakesef, but see Taz (6) and Peri Chadash are stringent, and see also Biur Ha-Gra.
Some write that this leniency is reserved for foods that were cooked on Shabbos, and does not apply to foods that are cooked during the week, and, coupled with the dispute over the leniency itself, one should preferably not rely on it. However, if there is nobody else to cook for you, there is room to compare the case to cooking on Shabbos, though there is room to consider whether the ability to buy ready food is equal to the possibility of a Jew cooking.
The reason why bishul akum might be more lenient than other rabbinic prohibitions, which are prohibited even for sick people (who are not in danger), is because the prohibition is not inherent to the food (Mishnah Berurah 328:63, quoting from Peri Megadim). See also Tzitz Eliezer Vol. 21, no. 61.