
Ultimately, I just gave up and bought kbdedit, and that seemed to do the trick.

I tried using the method indicated in the webpage I linked above, but couldn't get it to work. The other program that I found, kbdedit, that allows you to do this, is locked behind a paywall. Normal keyboard layout editors, such as Microsoft Keyboard Layout Creator 1.4 do not allow you to edit VK_KANA, which is the virtual key code that controls kana input. This webpage explains how you can use Visual Studio and Windows Driver Kit to edit a registry file called kbd106.dll to make japanese input compatible with ANSI. There is one way that I believe I can fix this. As such, when I try using kana input, some kana keys that take the place of numbers normally are in the wrong place. I have configured my custom keyboard so that punctuation and numbers are in a different place.

Unfortunately, this solution gets rid of our colemak keyboard for inputting romaji. If you have this problem on previous versions of windows, you should go into the registry edit and under HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\Keyboard Layouts\00000411, double click "Layout File" and change the data to KBDJPN.DLL.

Previously, I had assigned a custom Colemak keyboard to this position, and while this works fine with romaji input, it seems that the Colemak.dll file cannot be used with kana input.
#Microsoft japanese keyboard layout .dll
dll file associated with the KBDJPN layout in the registry editor. The kana input started to work after I updated. I believe I have found the root of the problem.
