Insisting that he needed to sleep,-- This is correct and is past indicative and past subjunctive. There is no difference in form.
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MaroldInsisting that he needed to sleep, Harry ...'insist' is not being used in the same sense here as it has when it's used with a subjunctive. Here it just means that Harry proclaimed quite strongly that he felt a need for sleep. He is not demanding of himself that he sleep!
Maroldwhen the past subjunctive is ever usedAfter 'if' and 'wish'. These are the two uses that immediately come to mind.
Marold"He insisted that he went to bed."Is the first 'he' the same as the second 'he', or are these two different people?
"He insisted that he go to bed."