Drapery panels on either side. Butt the right hand rod right up against the fireplace wall/brick with no finial. The other side hang rod and finial as normal. This is the standard way to hang curtains where one side is an issue. It disappears and you will not notice it. Place draperies and rods half way between the ceiling and the top of the window molding, or as close as you can if using pre made drapes. BTW, Ikea has longer lengths if needed. You should hang the drapes as you would on the sliding door, but mount the rods and brackets well outside the opening, so that they just cover the sides.

I wouldn't do draperies in that space because of the placement of the fireplace, but an inside mount cellular shade. You could get one that is not opaque if you never need room darkening. It will control the light without fully blocking it.

I'm assuming no treatment necessary on the sliding door?

Thabks for you suggestions! I'm not sure drapes would work as the hearth protrudes out on the sides, and would be in the way of the drapes. I should've also mentioned that inside mount shades won't work either, since the window frame has a lot of extra trim inside, making mounting anything there impossible.
Outside-mount shades might work.
Originally, there were wooden plantation shutters mounted inside the window frame, but they made the room feel very heavy and severe. Hoping for something lightweight and modern.

If you just need light control, what about a sun shade mounted outside above the window? They are remote controlled, don't block the view but do block the sun while allowing light into your room.

I got mine through Costco and love it. It is long enough that it could be mounted up under the eaves and still cover the window when extended, so it's not really visible from the outside and is protected from the weather (although they are weatherproof).

Sheer fabric shades like this would be nice:

Hunter Douglas Sihouette Sheers · More Info

Here's a closeup:

Hunter Douglas Alustra® · More Info

These allow light to filter in but retract completely when not needed. They tilt so you can adjust them to allow as much light in as you wish. They are very soft looking and warm up the room without bulky drapery.

Thank you for your suggestions!

I am loving the idea of sheer fabric shades - I think the soft light they will create will help make the room cozy. We were considering solar shades, but they feel a lot more sterile and industrial, almost. Really craving a homey feeling for this family room.

And thank you for the suggestion to put some artwork on the wall behind the TV - will definitely do that!

Vertical blinds for BOTH the window and the patio doors. They come in vinyl (least expensive and easiest to keep clean), fabric, faux wood and wood. The vinyl come in a lot of different colors. Clearly you like white -- which is an available color -- but you might also consider a tan or pale blue, depending upon any color changes you intend to make to the room in the immediate or foreseeable future.

https://www.google.com/search?q=White+vinyl+vertical+blinds+both+window+and+adjacent+patio+door&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjnkMOlw_flAhUyVt8KHSRACCgQ_AUoAnoECA4QBA&biw=1328&bih=617#imgrc=oABH_eKAx1rDJM:&spf=1574209483030

Would not do drapery panels if the fireplace is used for heating -- wood or gas -- or unless and until you create a less cluttered environment by investing in some dual purpose furnishings -- storage plus an additional use.

A pair of wheeled storage ottoman with flip tops that could be used as extra seating or table tops might be worth considering.

You might consider investing in foot lockers / dorm trunks: A pair of square foot lockers could be used as end tables and/or bed side tables. A rectangle foot locker could be used as a coffee table, leaving the table you have free for a play surface. Avoid the plastic kind of foot lockers and dorm trunks; choose the sturdier kind that could withstand a child using it as a seat.

https://www.google.com/search?biw=1328&bih=617&tbm=isch&sa=1&ei=C4rUXZWEB9KJ5wK3mqaIDA&q=foot+lockers+for+dorms&oq=foot+lockers+for+dorms&gs_l=img.3...4884.12351..12532...0.0..0.231.843.9j0j1......0....1..gws-wiz-img.......0i10j0i30j0i5i30j0i10i24.zkh1MhBdCCY&ved=0ahUKEwjVtYC7xfflAhXSxFkKHTeNCcEQ4dUDCAY&uact=5#spf=1574210076087

Please no vertical blinds. just no.

Not a pro.

Though even I would not put vertical blinds in a formal living room filled with upscale furnishings, hanging vertical blinds in a family room with a much more casual vibe, such as one obviously used as a children's play room, are very practical. Have had them in a downstairs den and would use them again for a space very much intended to be informal.

So here is the deal: if you want to do window treatments on your window and the sliding doors, you can go with a different operating system but the same fabric. The window treatment for the door requires a panel system or a vertical blinds system (not the old style you remember from your first rental) because it should operate in the same direction as your door for easy opening and closing. Having this said, the option for your windows are roller shades, panel blinds or vertical blinds in the same fabric. Don't compromise the functionality over the design...they must go hand in hand.

Roller Shades, Panel Blinds and Vertical Blinds REINVENTED · More Info

Thanks for everyone's suggestions! Ended up ordering a pleated fabric blind in a light khaki color for the window. Still on the fence whether to put anything over the sliding doors as we quite enjoy the view into the yard.