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Once I get this figured out, my objective is to use a gradient fill across the entire logo. The tutorial I'm using says to use Path > Outline Stroke. I can get gradients to work fine for each individual letter, but how do I get the gradient to work (smoothly) across the entire logo?
You may select everything, Object>Compound Path Make, and then apply the gradient.
It seems that you have joined the lower parts of the k, and I believe you can just use the Scissors Tool on both the lower corners and delete the horizontal path in between: it looks like Miter Joins, not Butt Caps meeting, as it would be if there were a separate independent horizontal path at the bottom.
Since ultimately the logo is going to be a single shape with a gradient, don't worry about connecting the paths of the individual letters. Your "before" example is just fine. Once all the individual paths are created, you can go to "Object/Path/Outline Stroke". If any of the original paths remain, you should select and delete them, leaving only the outlined shapes.Then click on the first icon in the Pathfinder palette (Unite). Since they're overlapping, they'll form one shape which can be assigned a gradient across the whole thing.
Thank you, thank you, thank you. Worked like a charm.