The question asked is the following: 
The book says that the correct answers are: \$V_{limit}\$ = 0V, \$V_{diode}\$ = 5V.
I don't get why it ended up like that. My intuition says that, "fails to open" means not functioning, and so no forward current flowing in the diode. In that case, the voltage across the resistor is 10V and the voltage across the diode is 0V.
Best Answer
No current is right, because your circuit is interrupted, and you need a closed circuit to get current flowing.
And when there's no current through the resistor there won't be a voltage drop across it neither, per Ohm's Law: Voltage = Current x Resistance. So 0 V across the resistor means the full 10 V is across the diode (not 5 V!).
If the diode would fail shorted, there wouldn't be a voltage drop across the diode, and the full 10 V would be across the resistor, giving (again due to Ohm) 10 V/ 1 kΩ = 10 mA.