Defending Irbil is one thing, but to actually retake territory, I'm not sure what can be done in such a limited fashion. The recent operations around Mosul, southwest of Irbil and in the Sinjar region seem to show that despite their vaunted reputation, the Iraqi Kurdish peshmerga, for various reasons including lack of equipment and pay, just aren't up to par with the Islamic State fighters, sometimes not even on their own or non-Sunni territory. The way the peshmerga fled from Sinjar was in fact quite galling, considering they had set up that place as a safe haven for refugees but when the Islamic State applied pressure to the area, the Peshmerga literally fled just as effectively as the ISF forces did. It was kind of a sobering display of a lack of martial prowess or will.
In order to turn this around, the US will have to move beyond just pinpoint strikes. They're going to have to invest heavily in air corridors to allow aid and supplies and munitions to arrive to supply the Peshmerga Army, regardless of what the Iraqi Central Government desires. The Iraqi government was supposed to share a significant portion of its budget to the Kurdish North and furthermore when the US left, the Kurdish segments of the Army were supposed to receive 200 million dollars worth of American equipment which again the Iraqi Central Government refused to release to them. The peshmerga as they are, seem completely incapable of an effective offensive action as of late, and have trouble even holding territory. At a minimum they're going to need lots of small arms, ammunition, rocket launchers, and artillery to defend themselves, and likely a fair number of vehicles to begin any sort of offensive action. They also need to be able to sell their oil on the market, circumventing the Iraqi Central Government, so that they can actually pay their peshmerga fighters. The Kurds don't want to sacrifice themselves to save other people solely fueled and funded by goodwill after all.
The airstrikes will likely have to be expanded too, and bring in more FAC's and other advisers to help coordinate airstrikes and the Peshmerga forces. I mean airstrikes without actual eyes on the ground are going to be of limited value against forces like ISIS. And that's just a truth of warfare. The air campaign against the Taliban back in 2001 was negligible to the point of being mock worthy until Green Berets and other Special Forces embedded with Northern Alliance fighters were able to directly target enemy forces with airstrikes.
The downside of supporting the Peshmerga is that it will help cement what is likely already over, which is the borderlines of the state of Iraq as we know it now. If this crisis ends with the Kurds more formidable and they attempt to become more independent of the Iraqi Central Government, the US fears are that the Shia governed portion of Iraq will be pushed further into the Iranian camp. But the Iraqi governments recalcitrance seems to have compelled the US to either largely ignore the crisis, or support the peshmerga in lieu of a lack of any other practical options.