How do you get a director of a limited company to resign?

You will have to examine the Articles of Incorporation and the By Laws to see if there is any provision for him being voted out, or for him being ejected for a rule violation.

Failing that, it may be that he has broke the laws of the state in which the company is chartered.

If you find any circumstance in which he can be terminated, then you can go to him and offer him the chance to resign. That's known as a "forced resignation", though it is the custom of the Western World to pretend that all resignations are purely voluntary, even when they blatantly are not.

On the other, you could always offer him a bribe, either in a one time payout, a retirement package, some non-voting stock options, or other things collectively known as a "golden parachute". It's the custom of the Western World to pretend that such payouts and payoffs are not bribes, but "rewards for long years of good and faithful service", even when it blatantly is not.