I'm looking to make the entire letter solid. I want to complete the grungy white space and fill it out with solid black. Is there a simple way to do this?
Only those holes, that 100% intersect with black fill, can be removed fast. But grunge on the edges can't be. There is only one way to achieve this with good quality – you need to draw path manually.
For the bulk of it....
Select only the outer path and that counter (hole) where the strokes cross. You'll have to use the Direct Selection Tool (white arrow). Just click-drag outside the shape until you barely touch the edge.
Object > Lock from the menu.
Still with the Direct Selection tool, click-drag across everything.
Hit the Delete key.
Object > Unlock All from the menu.
That will remove all the internal paths leaving the overall shape. It will not do anything to alter the outer edge, but it will "fill in" all the internal holes by removing enclosed compound paths.
The outer edge would need to be redrawn or manually corrected.
A quick and dirty (not perfect) method is to bring in the image into Photoshop at high resolution (black and white works best). Then apply a good amount gaussian blur to it. Then, apply a Threshold adjustment layer and play with the level. It will make it solid, but you will lose some detail. Then you can use the Burn or Dodge tool on the areas of the font you want to make thinner or heavier.