Iago does relish the death bed scene. His plans of attack did work against Othello: Othello murdered his wife and himself. Ironically, if it had not been for Emilia's outspokenness, Iago would have triumphed entirely. But, I do think that Iago wanted to get caught; he wanted Othello, before he committed suicide, to know it was he who destroyed him. I think he wanted to kill his wife as well, and getting caught was enough justification for doing so.
Most of all, I think Iago relished lying in the bloody bed. His psychological goal was to turn Othello's love (eros) and marriage bed into a death bed (theros). He succeeded magnificently, and his wounding was well worth it (he is a soldier after all).
That Iago fails to speak at the end is ironic and fitting. He has nothing left to say, which is to say that he has said enough (his words were weapons, and he has discharged them all). The devil tells no tales. To do so is to weaken his motives in the first place. The premise of the play deals with gender and language: he who controls words controls others. Iago won most of the battles and probably the war. Now that it is over, and he has been captured, he will not give his captors the victory in a confession.
"I bleed, sir, but am not killed." In these words, Iago seems triumphant. Evil incarnate (Iago) can be wounded, but not eradicated. Iago's last words are
Demand me nothing. What you know, you know.
From this time forth I never will speak word.
Again, Iago seems to be victorious. No torture or persuasion will make him talk. But, if you look more closely at how the play ends, we see Iago as anything but victorious. Sure, he accomplished his goals of detroying Othello's marriage and his happiness. But each of the main characters slowly disentangles himself from Iago's influence. Even Roderigo recognizes Iago as a "damned, inhuman dog." Iago never taints Desdemona's love for Othello, even when Othello takes her life. Desdemona dies trying to cover for her husband. Her love remains strong even as she is dying. Othello's last speech is devoid of remarks about Iago. Honorably , Othello takes full responsibility for Desdemona's death and serves as his own judge, jury, and excutioner. He is no longer Iago's puppet, but the noble man he once was. Even Emilia refuses to side with her husband as she chooses to tell the truth rather than obey Iago's injunctions to go home. She dies outing her husband.
Iago who was so clever with his words must in the last scene resort to action--stabbing his wife--in his desperate attempt to contain the damage to him that Emilia has caused. Iago can speak no more because Emilia has told the truth about the handkerchief. There is nothing more he can say to defend himself. Shakespeare refuses to grant this character a confession or an expression of regret. His soul will not be saved, and he will be punished in this world and the next for what he has done.
In the end, even Iago must realize that has failed. He has not "plumed up his will" as he intended to do in Act 1, will not take Cassio's place, and does not destroy the love between Desdemona and Othello.