The ring-out challenge in the Musphelheim crucible is one of the more challenging ones in that the enemies are pretty much impossible to kill using standard attacks. Their health regenerates so quickly that no matter how much you throw at them, they fully recharge their health during the smallest gap between attacks.
The secret to this challenge is not to use attacks at all. Focus on using the nearby lava. If an enemy falls into the lava, they will die instantly. So rather than trying to unleash hell on them through attacks, Use the heavy attack (R2) using the leviathan axe to shoot the enemy up into the air. You can then use standard attacks to direct them toward any nearby lava pits.
You are best off standing near the lava pits and letting the enemies come to you. It is a lot easier to let them get into position so you can use minimal attacks to knock them into the lava.
If you need some additional help or tips with the Ring out the challenge in Musphelheim, checkout the video below.