what kind of rug would you put here

Tape out what is feasible, but given such a high traffic area and its proximity to stairs, I’d skip. If you want to add interest, consider adding art and also create an attractive vignette with a larger lamp and eyes will be drawn there.

Why a rug at all?

I would skip an area rug there, because of the staircase and the irregular shape of the hallway.

If you must have "carpet" there, then consider having a broadloom cut and bound to fit the shape of the hallway with a few inches of hard flooring exposed all the way around.

Like this, cut to the shape of the space/room with exposed edge. It usually gets tacked down to eliminate sliding/tripping.

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a few more photos would be helpful.

I would move the chest to the wall where the framed diplomas are, and hang a long tall piece of art at the top of the steps. no rug, unless you can fit a runner.

It isn't 5 x 14. It a jog and turn, two separate spaces at most, and each would need a small area rug, matching OR complementary.

What are the feet and inches....separately please...and there is no need for one at stair landing top, unless you relocate the drawer chest. It will be very awkward.

Seems 2 @ 3' x 5' or max 2 @ 4' x 6"