Just eyeballing it, it looks like you could use a 4x6 (with the 6" dimension running left-to-right in this picture) as a secondary newel post for the top of the stairs as you originally propose. By using a 4x6, the new post would butt up against the existing one, making one solid post. This would narrow the entrance to the top of the stairs, though, so you'd have to measure and ensure that the remaining opening still meets code for safe ingress/egress, especially in emergency situations.
To attach this, you should remove the deck board there and see what's underneath. You'll want to attach this new post to the deck framing if at all possible to ensure it's sturdy, as people will lean on it, some will use it to pull themselves up the last step, children will play on it, etc. If you attempt to attach the post to the top of the deck board it will wobble from day one and only get worse over time, no matter what type of fastener system you use. You want the new post to be just as strong as the rest of your railing system. Cut that deck board to a shorter length (as necessary) and refit it when you're done installing the new post.
Once the post is in place, you should be able to replace the top rail with a longer piece to cover this one. You may want to put in a new bottom rail, just to complete the look, though it would be essentially non-functional and technically unnecessary. (I believe code calls for a maximum 4" gap in railings to prevent children from getting their heads stuck - yours looks like it's no more than about 2", but I'm eyeballing from the other side of the internet, you should measure to be sure.)
On the post at the top of the stairs, I'd suggest installing the gate to hit below the current hand railing. There are also code requirements about railing heights on stairs, and your current railing probably meets that. Moving the railing may move it out of the code-acceptable range. You could build a taller gate with a notch out of this corner to clear the railing (more complex to engineer, measure, cut, & ensure it's all square and looks nice), or just build a shorter gate that clears the hand rail and its mount in the first place. NOTE: You should check code for suitability of a shorter gate - the height of the gate may have to match the height of the railing, or at least be taller than the hand rail on the stairs, necessitating a notched gate to work around that railing.