You have a lot of potential here. The house is a very open one and I must say, you have too many colors going on that don't meld with one another. You have a pool and are on a lake, so the house needs to feel like you are bringing the outdoors in. Here are some basic suggestions:

1) Consider painting the main rooms in this area one shade. It will feel more like the outdoors are coming into the house. I would make it somewhat neutral and that does not outshine nature. I like the green in the sunroom, but you may tire of that before long. Choose something more neutral like Behr Perfect Taupe. It tends to change its tone during the day and is the perfect greige color. Paint the ceiling in the den the same color as walls. You can use color in your accents and have more paint color fun in the bedrooms and baths. I would not do an accent wall in the den.

2) Paint the trim in the house one color so there is cohesiveness from room to room. I would use the white trim that is already in the sun room. That looks fabulous. Paint the celing trim as well. The fireplace you can leave dark, since it should be a showpiece or later choose a specific color.

3) I like the idea of replacing the stained glass with windows. Is that a step-up under those windows? If so, how about window seats all along those walls under the windows?

4) You have too much furniture in the room. Place the couch where the 2 large chairs are. Place the 2 large chairs where the couch is. Remove the rest of the furniture. Use a large square coffee table in the center.

5) I am not a flooring expert, so will leave that to the flooring people. I do not see your current floor as offensive.

After that is done, there is plenty that can be done with accents. Choose accent colors, and accent with pillows and throws. Use a few large impact accents rather than a few small ones. One example: the clock is way too small for where it is and gets lost.

The wood beams are very distracting and lift the eye up. If you paint them the same as the ceiling it will make the beams go away. Replace or spray paint the ceiling fan so it is also white.
Since you are in Florida, I would stay with your tropical colors but your furniture doesn't fit with the tropical pastels. Since the room is dark you could consider peach since and take that down the hallway to the other rooms.

Thank you all for your suggestions. I appreciate your opinions!

You don't need to paint the trim a different color these days...paint it all one color baseboards and all. Check Farrow & Ball paint colors and do the entire area the same color. Create a clean palette to work with. I would white wash the beams. The room is too concentrated on TV viewing...I would move the furniture around so that it looks more conducive to conversation and not a movie theater set up. Buy the largest area rug you can find. I would go with sisal. Paint or change out the ceiling fan...and don't buy one with a light fixture. Move clock. Get a central basket for magazines.

Put French doors at all the openings across the back of the house and paint them and the beams and trim a light shade to look more like this pic.

I would keep the colors neutral so it doesn't clash with the adjacent rooms, and add skylights. The difference will be immense. If you're in a hot area, get operable skylights so you can open them in summer time and get a breeze through the room. Upgrading the windows and doors will also help with the look of the room.

I think you must start with the floor plan also. I suggest all mouldings white, add mouldings around all doorways to have continuity as you have some with and without. Either remove all doorway mouldings or add those that are missing. I just painted stained oak mouldings to update my bathroom. Kinda hated to do that, but wanted to update. Paint the beams and ceiling off white.

I think you have the two lounge chairs together, facing the TV, because you as the owners .... they seem to be your favorite chairs. Pull the two lounge chairs closer to the wall, meeting at an L-shape with the sofa. Place the two lounges and the sofa so that arms are nearly touching each other. Tighten up the grouping. The best would be to have the sofa face the Tv and two lounge chairs where sofa is.

With that said, remove the green chair and end table that are at the left of the fireplace. A bit too much furniture. That should be a walkway to get to the other rooms. Use the green chair in place of the chair facing the kitchen but not centered in front of the TV. Angle it.

For extra seating, instead of different chairs. Add some ottomans centered in front of the fireplace. Maybe some floor pillows if it doesn't get overcrowded.

Sofa stays opposite the fireplace with coffee table. End table next to sofa where basket is now. End table between two chairs. Lamps on tables. Haven't found a place for the floor lamp yet.

Try moving the furniture for some of my ideas to see what you think. I think you will see and understand how the room will feel, more comfortable and better access to the the other rooms.

Post some pics with the changes and we can go from there. Similar idea below.