This is the file and its path that Kate's spellchecker uses to hold all of the words that you "Add to Dictionary":

C:\Users\<your Windows log-in name goes here>\AppData\Roaming\en.pws

Modify the path below to show your Windows log-in name. Then, select, copy, and paste that modified path into the "File name" box of Kate's "Open File" dialog box and click "Open." Finally, scroll down to find the file "en.pws" and select it and click "Open." You can then use Kate to edit the file, for example, to remove words that might have got in by accidentally clicking on the "Add to Dictionary" button when spellchecking:

C:\Users\<your Windows log-in name goes here>\AppData\Roaming

Note: This is the situation for Windows 7. Variations of this situation might be the case for other versions of Windows.