I know that I'm pointing out the obvious, but it's still worth discussing.
Jay using Sun's hand and knife to slit his own wrist is NOT a metaphorical "see how it feels when you watch the love of your life get hurt?" thing, but a more sinister and more transactional act that works in Sun's logic—or at least, how Jay perceives to work in Sun's logic.
Sun lives for revenge and debt: he blames himself for the death of his family, lost his dignity to pay all of his debt, and saw himself as disposable for the sake of revenge. Living doesn't matter. Paying back does. He's more than willing to die for as long as he's clear of his debt and his enemies die with him.
So, Jay puts it upon himself to make Sun hurt him. That way, Sun will have to compensate him, be indebted to him, to need to pay him back. He literally imprints the idea that Sun tried to kill him, so from now on, he has to keep an eye on Jay only.
