We need pics of all four walls to make sense of the space, I think. I'm not getting a great sense of the whole room. But top of head I'd say turn it into a great deskwork/homework (?) area, and make it gorgeous by having very organised storage space so you don't get mucky piles of paper (guilty, M'lord). You're right; it's much too nice a room to waste by just walking through it.

Think art, perhaps, instead of the mirror. Something you and your whole family really love, to greet you and your visitors. If you want a little check-up mirror, put it to the left of the front door.

That rug doesn't work on that floor, for me. But I'd be loth to suggest anything specific without more general thinking and planning. I just think a plain rug would work a lot better. And bigger.

There seems to be too many openings and access points to have any proper furniture setup in this space.

i wouldn't have a lot of small items of furniture.
Where the desk is I would have a large armoire for storage(toys if this area were designated as a playroom. It's an easy and quick fix for putting away any items that are hanging around )
A small bench under the window.
And under the mirror a closed cabinet with drawers (to hold keys, eye glasses etc)
ref pic

As was mentioned previously, I need to see all the 4 walls of this space. I also need to know what rooms are off of this one and how traffic flows through here. Off the top of my head, I was thinking of another direction all together. How about putting a round rug in the middle of that floor with a round table on it? From there, you could have flowers, seasonal décor, etc on top of it. I would make one of the short walls into a tall storage unit (anchored securely) with doors to keep boots, backpacks etc. The other walls would serve only to hold artwork. What is that white bench by the front door? Can you put a window seat cushion on it?
Let me know if you have any questions.

Very good advice from groveraxle. Couldn't say it better myself. Also agree with Anne Dee that there are too many small items. One suggestion might be...since it is used as a homework center is to get a rolltop desk and that way you can quickly roll the lid down to cover the mess when guests come over.