How can I repair a poor drywall taping job?
The whole wall doesn't need a skim coat. That patch just needs touch-up spackle, sanded correctly.

The picture does a good job highlighting what you failed to do: bevel any divots or ridges with a sanding sponge during the final sanding. This is not something you can do easily with a pole sander; it'd be easier with just the paper in your hand if you've no sanding sponge. Use a swirling motion to reduce any deviation to a nominal profile. Fudge it.

It doesn't have to be flat, it just has to not have any ridges that will catch light and show the imperfection. Keep this in mind when sanding the extra coat you're going to put on, to fill these giant voids, that are next to the ridges that are covered in paint and are at this point basically unsandable.

Mudding an optically flat wall is beyond most people's abilities (if not, the general realm of possibility). The closet you're going to get (or ever really need to) is 'looks' flat: nominal. 

If you can feel a 'catch' with your fingers, you're not done sanding yet.