How can I install a stair skirt to fit the stairs instead of running alongside?
I think this may work for you:

This article, "Scribing Skirt Boards" by NORM YEAGER, provides a pretty good explanation of a technique to install a stair skirt "on top" of an existing stairway. You might want to do a couple of "practice runs" before you do the actual skirt...depending on your skill level and the steadiness of your nerves, etc. It isn't difficult, but it is one of those things that gets easier with a bit of practice.

What they're detailing here is laying a 1x12 across the top of the stairs and scribing the outline of the steps onto the board, for cutting out the profile of the steps and then installing it above the steps instead of between the staircase and the wall.

Personally, I think that I'd use a pencil instead of a sharpened brad to mark the cut lines, but both methods should work.

If there is an existing skirt, you could install your new skirt over the top of it to effectively make the two skirts into one thick enough to cover the gaps. 

If there isn't an existing skirt, making one out of 1x12 might be thick enough to cover the gaps, but you may want to use a 2x12 to make the skirt so that you'd have the additional thickness to cover the gaps.

Depending on the surrounding woodwork, etc. you could augment the existing molding to match and blend in the additional thickness of the skirt.