I think the problem is actually not so much the shape of the room but the fact that at the "pitch of the roof" part you have a large full length window and fireplace. Essentially, because of all this there is no wall to put your back against or place furniture against. My rectangular room has similar issues with a corner window facing the fireplace. We just put the kids toys there on the floor and low kid size stuff like a toy box. You can't put any large pieces there. So, I say, start by completely ignore the top triangle of the room. Its essentially dead or unusable space, but will add to the feelings of space and light in the room. Arrange your large furniture pieces in the bottom square portion, around a large square rug perhaps that would define the space. It might feel better that way. Then arrange your other bits around that. I think if you try to work the square it will be way easier to organise the room. Emphasising a square layout will feel more comfortable. Plants in that light bright corner was a good suggestion, they soften the angles. Hope that helps!

I'd be inclined to move couch & armchair along wall with large recess if they will both fit. Mount TV onto flat wall opposite. You could have a low unit to house electronics, or use the one you have. Have purpose-built storage cabinet w/shelving built into smaller recess. Some of the shelving portion could be backed w/mirror to help throw light into room. Use the shelves for books, objects of art, photos etc., and the drawers or cabinet below to store things you don't want on show all the time. I also think a large area rug will ground the family room area from dining/kitchen. A neutral one perhaps or you can inject some colour. Large art pieces on wall as mentioned. A largish plant in a raised pot would add some greenery. You may find you need a floor lamp near couch & chair. Regarding the kitchen, I think you might be able to fit a narrow timber sideboard at end of island bench so you have some additional storage there. I say timber because it would warm up the coolness of kitchen. I agree undercabinet lights good idea for additional task lighting. You could either go mirror backsplash or some gorgeous tiles. As there are a several changes you want to achieve in this space, it might help to do it in stages. Storage seems to be a priority (fam rm & kitchen) and rearrangement of furniture & TV. I would tackle this first. Additional decor ie rug, art pieces on wall(s) could be addressed next. Then splashback and some additional lighting if budget allows.

I am just wondering whether it is a good thing or not to go with solid colours like the dark blue (prussian blue), grey floors and white on other walls all in tiles. I see that a lot of designers seem to do a pattern on one wall, but I haven't found one suitable and whatever I do, I just don't want it to look out of date in 5 years time. I am not sure how about much of the prussian blue tile is appropriate in order for it to be noticeable but not too overpowering.

The hallway is bright and welcoming. Forget a mat as you have dogs as well as elderly family. Unless the chair is serving a purpose, I would remove it and add a full-length wall hung mirror which will reflect light and make the hall look an indeterminate length and you can used it to see your reflection before you leave the house.