|This is how my image looks like when I scan it. Nasty, eh? The ink has become more of a dull grey in areas, and you can see my brush strokes. I don't even like showing you this version, but it's for educational purposes. ^_^|
Open up the Curves box (Ctrl+M, or Image>Adjust>Curves). The two dropper tools on the right side of the box are the ones that you need to concern yourself with. First, select the dropper tool that's filled with black. Then just click it on a medium greyish area in your scan. All other greys of that colour or similar will turn black. Then grab the tool with the white ink, and click that into some negative space, and anything sort of white will now be a billiant white. Then select okay. Now you move on to the next step.
As you can see, the image is already looking a heck of a lot nicer, but we still have a small problem. The areas circled in red aren't quite black yet. All you have to do is zoom in a little and repeat the curves actions over again. This is the last step. Just be careful about which pixels you select with the black ink, and which ones you select with the white. Sometimes hitting a grey pixel in the black areas will make your white areas more grey, and vice versa. There is a before and after image below.