How to look after a stray cat?

So a few days ago we had a little kitten show up on our door. She seemed to have been lost from her mum or her family and was kinda scared which was natural. She was meowing like crazy and coming from a kitten as small as her, it really melted us. We gave her milk and tried to calm her all we could. She now lives at our terrace (we have a flat terrace) during the day and goes out at nights. I am fine with it as our mom is not much of a pet person and that’s the closest she can allow an animal to be. We go and see her every day and she trusts us now and loves to be around us. My concern is that for a couple of days, she hasn’t been eating at all. As a family that is vegetarian, we try and give her milk and mashed potatoes but I know that’s not her regular diet. I want her to learn to hunt for herself but probably she never got enough time with her mum to learn it. Is it possible that she has been hunting at nights? Is she sick? I would like to hear from those who have a cat on how I can help her or what her behaviour suggests. We can’t take her in as a pet and want her to learn to hunt yet until she does that, what is something we could do for her to be in good health?