If you have open work rights (also available through marriage or PR status), you don’t need a work permit. The illegality of kindergarten English classes breaks down like this:
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It’s illegal for a foreigner who doesn’t have open work rights, because it’s impossible to get a work permit for the job, as far as anyone knows. Max. penalty under the Employment Service Act: fine + deportation.
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It’s illegal for anyone (foreign or local) to work as an “educare provider” without proper qualifications. Max. penalty under the Early Childhood Education & Care Act: fine.
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It’s illegal for a kindergarten to have foreign language classes (whether English or Swahili, taught by a foreigner or not). Max. penalty under the ECECA: fine + being shut down (iirc).
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It’s illegal to run an unregistered de facto kindergarten. Max. penalty under the ECECA: fine + being shut down (iirc).