I would put a simple window film on the arched window. It will give you enough privacy, light coming through, and cover the bad view. You can use a slightly frosted one, or maybe one that looks like seeded glass to match the style of your home. There are los of options. For the square window, you can use a woven wood shade or other blind. For interior mount, you need to measure using the larger outside frame and look for one that will work with the narrow dimension for installation. I think it would be best to mount a roman shade over the window at the ceiling. Then you get the most light while the shade is up and don't have to drill into the beautiful wood frame. You could also mount a rod above the window and use a simple linen drape on each side to pull closed when you want privacy.

I would do some blinds on the bottom and leave the arch since this is a play room make sure the blinds have no cords.

It looks like a really lovely house with lots of light. There are several options for how to cover the arched windows. It is going to come down to personal taste and whether or not you feel the top of the arch should be covered. Personally, if the light isn't too much coming in, I'd cover just the lower portion of the windows and leave the arch exposed. I found some pictures of different options. This first one reminded me the most of your room. The curtain rod cuts the window before the radius of the arch starts. The contrast of the curtain rod against the white is not too stark and the curtains blend with the wall colour. I think the drapes colour matching the walls and trim helps it all feel like it belongs and doesn't leave a harsh line where the upper arch is uncovered. Similar idea here although the curtains shown appear to be photoshopped and aren't nearly full enough. I think the picture above, with lighter, airier looking curtains helps them feel like they just blend into the room, not accenting the window area. You could get a radius blind to cover the arch and then dress the lower part of the windows however you please. This picture shows a blind below as well. These curved curtain rods make me think of eyebrows....but if they were in a lighter colour closer to the colour of the wall and window trim they might be better. This type of thing can also be achieved with velcro attached to the window trim and the fabric attached to the velcro. This treatment covers the whole window including the arch. Those are going to be some $$$ custom curtains. You've certainly got the height in your room to achieve this look if it appeals to you. I'd have the curtain rods match the wall colour, rather than contrast based on the pictures you've shown us. Hopefully there is a good drapery/blind store nearby and you could visit them and ask for some help and ideas. If it were my room and the furnishings that you show in the second picture you posted, I'd choose the very first picture I posted above. I think it will look gorgeous with what you've got.

@Ely M Beautiful inside too! Gorgeous floors. Love those windows & would be a shame to hide any of the features of them on inside. I’m guessing windows are centered on the wall - looks off in picture but guessing that’s angle. I’ve never seen angled rods as @threers suggests. I also think any inside mount blinds would be custom & $$$$ due to shape & I don’t think outside mount fits these windows. I would go with drapes with the rod just above highest point on center window & with panels that split in middle & pull half to each side. I see the vent at lower right so you may need to use shorter length that ends just below sill. Even though the other window just like this one is evidently in another room, if you want to cover it at night, use same kind of drapes so the view is the same from street & the back side of drapes should be white. BTW, love the hanging light too!