You have an enclosed stair run, not really a knee wall. An easier way to handle the newel post is the bring it to the outside of the run, like below. Then you can anchor it from the front into the side of the run. Then the hand rail from that to the wall stabilizes the top of the newel.

In my limited experience, the stair stringer is the structural component, not the knee wall. I would think you could shorten the knee wall and install the newel post into the subfloor. If it cannot go below the subfloor and attach to the house structure, there is newel post hardware that they used to sell at home depot that can attach posts to subfloor. You might want to reinforce parts of the newel post by attaching it to the shortened knee wall, then you would have to trim out any part of the post that looks unfinished.

I think the main rule is that it needs to support a certain weight limit In all directions. You can add a lot of different types of strong metal hardware to reinforce the connection. You may even be able to drywall the knee wall around the post if the dimensions are within 3.5“ or 4”.

I would look at connecting the post to the subfloor using the kit, or through the subfloor if you can, and also connect the base of the post to the kneewall. Keep in mind the kneewall at the lowest point may not be very strong. But you can always attach more short 2x4s to the floor if you need to build up support.

It would help to know more about the post, like a picture with measurements.

I would avoid mitering the bottom of the post if possible. It’s not hard to cut back the kneewall if there is nothing important under it.

I assume the framing in the kneewall is 2x4, correct? 3.5” frame plus 1/2” drywall is 4”, plus the stair trim which does not look 2” thick (more like 3/4”, but I would not touch that part, and if it’s the stringer, definitely don’t cut or stress the stringer).

The above is based on personal experience. I am not a carpenter or structural engineer, I just had a very similar problem in my old house and I worked it out (with the advice of a carpenter). There was not a simple solution - I just had to work through each issue. But I was picky about the trim.

If you can go under the stairs to see how they are built, that would help. The stairs should be structurally independent of the kneewall. If they are not, the above may not apply.

It appears you have no inspiration.

Just loaded the photo…

but here is is!

Now there's some inspiration.

I think it would look and function better if the newel posts sat on the floor.

Why do you want them on the side wall? To save space?

If you have access to the joists below, you could anchor it below the floor, otherwise bracing it from both directions top and bottom. Railings like this are not an easy DIY even for experienced handymen. We hired someone for ours. There are so many angles and technical cuts.

Having them angled on the top of the wall like that looks awkward to me. It's not what is usually done. I'd do some searching for closed stairway newel post and make sure you really would prefer this look.

This is a way you can put the newel on or into the floor and trim around it so it looks like it is resting on top of the wall.

If you want to pull the newel back beside the first step, cut back the wall and replace that first vertical stud so that it is taller. Then you will have room for the newel beside the first tread. (Assuming the wall does not affect the integrity of the staircase.)

So if we decide to do something like this where the newel is notched to but up against the bottom of stair wall and sit on the floor, does it matter if we use a solid newel or one that is hollow with an insert that gets screwed into the floor?

You need to make sure the post and handrail will support the required weight. This is more complicated than screwing the post into a 3/4” thick subfloor. If you google, you can read about different ways newel posts are supported. This is a safety issue in case someone falls and grabs the handrail for support, or leans over the handrail.

The installation instructions for the newel post might help.