Sir! Intel states that the Inhuman Terrigen Cloud sterilizes and causes debilitating poxes upon the mutants.
We also have found out certain Inhumans die from the gas cloud as they have inferior genes and it is possible for them to die/mutate into monsters after being exposed the second time.
We also have parts left over from Nimrod as Lord Stryker used it to defend ourselves against the accused X-Men and their agent of chaos codenamed "Storm".
Nimrod could copy some of the minor powers of the vile mutants so we could use the leftover imprint of that witch Storm sway the gas cloud in a specific direction.
We also know that as appeasement to the mutants, the Inhumans plan to round up every mutant that they can find in the gas cloud area. However it has been shown that they cannot carry every single one in a given area. We know that the Inhumans drop them off at the closest area not affected by the gas then leave as soon as possible.
I propose a stinger attack in which we manipulate the gas cloud via Nimrod's parts smoke out the mutants in a given area. Those who are affected by the pox are sterilized (preventing their hellspawn seed from spreading) and soon die of the sickness. Most areas are registering those affected by the pox in their medical systems so by infiltrating the hospitals/hacking their mainframes we will be able to monitor the weakened mutant abominations.
The Inhumans will unwittingly help us purge the survivors as we monitor their movements and lie in wait when the drop off the mutants they claim to have saved. I would advise not attacking the Inhumans directly as we do not need to fight a war on two fronts. As you said Mutants first, Inhumans later.
Soon Earth will become uninhabitable to mutants so they will inevitably leave the planet or succumb to the sickness. With the new World Shield program in the news this also prevents them from coming back to Earth anytime soon.