When I had a long, barrel ceilings front hallway with little natural light, I chose eggshell. It offered a glow that would have been more shiny had I gone with semi-gloss.

Semi-gloss is for durability like cabinets, trim and high activity areas.

You can reflect light with glass framed artwork, mirrors and keeping your decor in lighter tones.

Ultimately you can't add natural light without a skylight or solar tube. You have to accept your homes position and learn to work with it.

Semi-gloss will make people think of a bathroom. One way to deal with low light situations is to choose a warm, bright wall color. BM Cotton Balls for instance. If this is too white for you, look at BM Seashell or Soft Chamois. I wouldn't go above eggshell or satin (different brands call it different things).

I helped a friend paint her condo and she wanted semi gloss on the walls. I was against the idea but it looked lovely. Gloss on trim as well. The only concern is the condition of the walls, semi gloss will highlight all imperfections. I would also consider painting the whole place the same way, if some rooms are semi and others on same floor are matte, for instance, it might look strange. I don’t know…

The answer to dim is electricity. The answer is decoration. Deep dark spaces can look larger, more inviting, cozier. Light or dark, all need ELECTRICITY, and a combination of task, ambient, accent lighting.

No to shiny walls. shiny trim? Fine.

IMO a matte finish for walls and better lighting is what you need to brighten a space not semi gloss walls that will not only reflect light in a bad way but show every tiny impefection in the walls and believe there will be many. There is nothing wrong with turning on lighting in a room that does not get enought natural light I like all LED 4000K for my lighting it gives the look of bright daylight . If you want real help here you need to post pics of the space

My open concept living room-dining area-kitchen faces north (that's where the major view is). When I bought the house, that entire area was painted in a medium dark color along with a very dark wood floor. The kitchen cabinets were dark oak and appliances were black. When I remodeled, I went with a light porcelain floor tile throughout, light maple cabinets, stainless steel appliances and light colored walls in a satin finish. The popcorn ceilings were stripped and painted flat white. This helped immensely to brighten the space, especially the living room and dining area. It was sufficient for these areas since the living room has a wall of windows and the small dining area has a patio door which takes up most of the outside wall. There is no need to have the lights on continuously.

The kitchen was a different story. While the lighter finishes helped, there's only one small window above the sink. A bedroom above the kitchen precludes adding a skylight or solar tube. Even though I don't care for nor require operating room light levels, the solution was, as Jan Moyer suggests below, electricity. Working with my electrician, we decided on some well placed recessed lights. I'm not a big fan of swiss cheese ceilings but the recessed lights were the best solution. The can lights are usually on from morning to bed time except when there's no one in the house. Being LEDs, my electric bill is still reasonable.

Never ever use a sheen on walls for the reasons given. You are right it's terrible and plastic looking. Here is my theory based on experience as we've done both:

The eye will focus on the glare off a shiny wall, but with a flat finish with no sheen, the eye cannot find a surface on which to focus. The brain then interprets the confusion of the eye as infinity, rather like looking into a thick daytime fog. That is the impression anyway. No matter how close or how far from the wall.

Light reflectance value is not dependent on sheen but whether the paint is light or dark.

But not too white for a North exposure, a stark white will look cold and gray in back corners. A warm, not yellow, off-white is the most popular choice.

The example of teal and charcoal colors above work with the dark North exposure, play it up, and make it cozier. But notice the flat finish. The painting in the teal room looks like it's floating in space right? I love that warm teal, look how even the dark corners of the bookcase are nice. The sun is coming that room but a daylight bulb as suggest will be the same effect.