Is it permissible, Islamically, for a Muslim living in Israel and working at a Jewish institute to visit an elderly Jewish colleague in the hospital and take care of him and his food?

It is permissible to show kindness, gentleness, benevolence, and give charity to a non-combative disbeliever, as long as it does not involve allegiance or heartfelt love for them. It is desirable that the intention behind showing kindness to them is to invite them to Islam, along with supplicating for their guidance. The evidence for this is the saying of Allah the Almighty: "Allah does not forbid you from those who do not fight you because of religion and do not expel you from your homes - from being righteous toward them and acting justly toward them. Indeed, Allah loves those who act justly" (Al-Mumtahanah: 8), and the action of the Prophet, peace be upon him, with the Jewish boy who used to serve him.