I found what I needed, but this info was VERY hard to find, as Googling for things like "resetting the state of the keyboard interrupt in x86" just got search results showing other people asking what I was asking. No really useful answers. Only one tiny hint I did find from Google (via reading an answer to someone's question) mentioned sending the EOI (end of interrupt) signal but didn't explain how to actually do that. It took a couple hours of research deep in programming tutorials to find the code I was missing. All that work just to find these 2 lines of code that I needed. That code being:
mov al,0x20
out 0x20,al
This sends the number 0x20 to I/O port 0x20 to reset the interrupt state to allow it to be used again. I don't know why Google couldn't have directly pointed me to this answer, but I guess Google isn't as smart as everybody thinks it is.
And I'm also kinda shocked, that on an ASM forum like this, for an assembler capable of writing 16bit code for DOS, that after all these hours the post has been up nobody here seemed to know the answer either (as nobody replied to my post, other than myself).