Once your broken, you're broken, you don't take any more damage of that type.
The rules say, regarding being broken in Empathy:
You break down in despair or self-pity. You must either explode in a violent outburst, kicking and breaking everything around you, or withdraw from everyone around you.
When an attribute score reaches zero, you are Broken. This means that you are put out of action in one way or another.
This means that the player has a choice, either to 'violent outburst' or to withdraw from everyone. One thing to remember is that the basic rule for being broken in general still applies:
When an attribute score reaches zero, you are Broken. This means that you are put out of action in one way or another.
Since you can't roll for any skills, the 'violent outburst' just means that the PC throwing a tantrum, but doesn't runaway. It is up to the GM to decide if the tantrum PC breaks anything nearby, like a tent pole or a small table, but he doesn't do anything useful. You are allowed to runaway and that is probably the better choice since you can't roll for any skills, so you can't defend yourself, but as soon as you have runaway, you can sit down and cry it all out until your friends come and get you.
Coup de grace can only be performed against characters that are defenseless, which is being broken in Strength or Agility (or technically being a asleep or bound to a chair, etc.). Being broken in Wits or Empathy, is not being defenseless. Sure, you aren't allowed to use any skills to defend yourself, but the character isn't exactly defenseless, he can still move around, etc. Perhaps not in a rational way, but still enough so you can't cut his throat.