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It probably the stem/cartridge which you access from the top of the sink. The stem is what the handle is connected to. You'll need the model to get the right part. If you don't know the model can you post a picture here. I might be able to identify it. You can post a pic here by clicking on ADD FILES next to the SEND button.
Looks like you removed the entire valve unit. You only need to replace the stem. You back off the hex just below the knurl where the handle attaches. Cold stem part number GP77006 hot side is GP77005.Here is a link to the parts blow uphttp://www.us.kohler.com/us/catalog/svg_viewer.jsp?productId=421471&diagram=2
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That's correct. The fix is to replace it. In order to do that you have to remove it from the valve body. It's not all one piece (what you show in your first pic). If you look at the parts link I gave you you'll see how it goes together.
If you look further down the same page you'll see the parts I'm talking about. Here's the link again
If you can't back it off with one wrench on the flats at the bottom of the valve body and the other on the stem hex then you can try soaking it in CLR or Limeaway. That might even free up the stem to turn (but I'd replace it anyway). If you still can't get it apart then you'll have to replace the entire assembly.