More pictures from interior.

Here is an interior shot from living room looking out.

Drapes. Sheers plus something heavy like velvet or damask. I'd personally skip the puddling on the floor look, but that is your call. There isn't enough room to do something outside, but it is perfectly reasonable to close off a dining room and solve the problem that way.

I looks like you don't have any room for plants What are the measurements of these outdoor areas?

Talk to the neighbor: I'd remove that fence because it's not doing any good. I can't think of any landscape solutions aside from trellises in front of the windows with evergreen vines and you don't have enough room for those.

Plants in pots?

A gate in an arched arbor to block off the potting shed?

And inside the dining rooms, install bottom-up shades you can pull as needed.

Are you expecting to tear up the bricks? The brick work looks nice.

It seems that anything you plant outside to block the dining room windows will also block the living room window.

I agree with others. The privacy solution is inside the dining room. Louvered shutters on the inside on the lower half of the windows?

If you wish to keep the existing fence, it would look much better if you removed the top horizontal board and replaced it with a 2 x 4, and then capped the whole thing off with a 2 x 6, pitched at an angle (like a roof) to drain rainwater. I understand that weathering wood is desired for it's perceived low maintenance, but truly, it would look vastly better (and last longer) if painted. It would be a 1-hour job with a roller, and probably an hour to clean it with bleach before painting.

I have seen English Ivy grown into a chain link fence so that it was a turned into a green fence. It must be sheared like a hedge if one wishes to keep it from becoming too thick. Such panels could be strategically added above the existing fence. As an alternative, there is fake English Ivy that is made specifically to be woven into a chain link fence for screening purposes. Last week, I happened to drive by just such a fence. While it generally looked like ivy, it also looked very "engineered" with an artificial aura about it. (I stopped to investigate in order to determine what it was. The advantage, of course, it that there is never any trimming or worry about if it is getting enough water and fertilizer. (I added a link below.)

Another idea that occurs is to add panels of outdoor fabric that is stretched by being laced to a tubular steel frame (like a trampoline is stretched within a frame, but with lacing instead of springs.) It works better to use clothesline wire, cable or similar metal instead of rope, bungee or plastic ties. None of the latter hold up against sunlight. One advantage of fabric is that it doesn't create oppressive darkness.

Here is a link that might be useful: Fake Ivy for Chain Link Fence

I might spend my $ on finding or having built something like this (the planters):

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You could have them built taller than the fence, if they are not attached (to comply with height limits). Not sure what plants could grow in that space - what light there might be, but some tallish grasses would add some softness and more privacy, but probably only with sun. Evergreen ferns might work, too, as they will grow in the shade.