It is more likely he is not dead. The American government should be less 'yay! Lets end the Kim Dynasty rot and hope for someone we like!' and more 'Get well soon Kim!'
Because if he isn't dead, you'd still want open diplomatic relations. A bit hard to do if he saw that you were jumping for joy at the prospect of him being dead. Also if his sister does take over, you'd want a smooth transition. Telling the world how glad you are that her brother died and hoping that the Kim dynasty is finished would also be embarrassing for future negotiations.
I agree, it would be bad overall if he was dead.
3 big issues come up, one potentially good.
1 - The world needs to re-negotiate and re-strategies how to handle a new dictator.
Its possible this could be better, but at least we know that the nukes arnt flying.
2 - A new dictator will want to prove to her people that she is strong and capable. This could mean new missiles and nukes flying about.
3 - People may not accept her and start burning things. This will be a lot of bad for now, but may force change.