nè are words. But there are no
ně. I tried to Google this but I don't find any references to it. It's obviously not by final, because
kě all exist. Is there a rule?
There’s no rule as to what tones can be used with which pronunciations but there is a finite list of words & pronunciations in MSM (Modern Standard Mandarin).
When you start getting into Chinese dialects and topolects any pronunciation is possible.
Just as a side note né can be found in 哪吒 (Nézhā) the name of a deity.
No,There are some rules in the relation between tones ans syllables which put restriction on which syllable may occur in which tone, for instance, you can not find "man" with the tone "55" or the first tone, but they are developed from ancient Chinese and complex, almost no one is interested in those rules.