The puppet king did get his air force: half a dozen decrepit Ki-43 Oscars. (End of the Beginning, pp. 169-70) His problem was finding pilots small enough to fit in the cockpit. They were probably still in Japan, learning how to fly, when the islands were liberated by U.S. forces. They were likely given a choice: Fly for Japan or else...
On p. 289, the briefing officer (probably the Squadron's Air Intelligence Officer) for Crosetti's squadron (likely VF-17) mentions "Our assets on Oahu". Want to bet that those were either Marine Raiders or even OSS teams sent in by submarine? No actual guerilla warfare, mind, but they were sent in to observe and report.
EDIT: the B-17s that were at Hickam or the flight of 12 that came in during the attack were in all likelihood wrecked. Either by Japanese action or destroyed to prevent capture.
Ki-43 Oscars huh? Figures they'd be army aircraft. Even if it was only six planes, that still counts in my book as an air force. Or at least in this case an air wing. That's kinda funny that the main problem is just finding pilots small enough to fit in the planes. Not even finding trainers or anything, just finding tiny Hawaiians.
Yeah definitely likely those bombers were scrapped or destroyed during the fighting for Oahu.
I'd bet those "assets" would be OSS agents inserted via submarine. If they're just there to report and observe, you likely wouldn't need marine raiders to do a job like that.