Can you add a sunlight (or two) to the roof of your porch instead?

I think some photographs would help, also a drawing of the space with measurements.

Is there anywhere to hang a mirror in the eat-in area that would reflect some light?

I have a north-facing kitchen window with a lot of trees. I also live in a northern climate where in the winter it's dusk or dark at around 4:00. That is one of the reasons why I chose white kitchen cabinets, light backsplash, natural oak floors, and lighter granite counters. You didn't describe your kitchen. What wall paint color do you have? Cabinet color? Backsplash? Counters?

I don't think a high gloss ceiling paint will do much, and furthermore, when you have the lights on at night, it will be glare-y.

Hmmm. I have a wood ceiling painted white gloss and we don't have a glare problem - that I've noticed anyway. I can't tell you if it will help or not but your question was "Can I have a gloss kitchen ceiling" and my answer, based solely on mine is, "Sure".

Do you have any images of your space? It might be helpful to see ways to add light to the space. I wonder if you could add one of those tube light that add natural light through angles and mirrors through your ceiling. Light torchiere lamps pointing upwards can sometimes add light. And one thing I did in my kitchen was put small mirrors inside my glass cabinets with the mirror facing upwards toward the ceiling. Any light on them would be reflected back up toward the ceiling. Reflective surfaces like tables or even reflective placemats can also help reflect light into the room. I am sure you have done this already but the brightest bulbs your fixtures can handle is another easy way to brighten the space. Bright rugs or really bright objects also deceive the eye into thinking light. A bright red, pink or yellow vase in the space can add color making a space seem brighter. ~boxer