Oooo, that's tricky. Give a reasonable deadline, then - a month? Six weeks? Could you live with that? Whatever you decide, make it clear that you won't extend it.
Can you be generous in other ways? Offer to help box things up, or hang onto things until he has a place for all his stuff? Can you afford to help him financially with a deposit (I imagine as the main support for 6 children, you can't!), or with the costs of moving. Anything you can do to try to maintain a reasonably civil relationship with him, but also shifting him out the door!