Do you hear any sound when you turn the flash on? The inverter makes sound due to magnetostrsiction of the transformer. If you don't hear that, you're not to first base.
Unfortunately, Fixing this requires some electronic troubleshooting skills, and also precautions due to the high voltage involved. You could get lucky and find a bad switch, or a broken wire, but it will likely take more than that. If you can find a schematic for it and have the skills, then give it a try.
Also, you should stop using alkaline batteries in your expensive equipment. NiMH have several advantages, some of those being cost-saving, but the biggest of all is that they don't leak, whereas alkaline are guaranteed to leak. It's not that it's impossible for NiMH to leak, but I've never seen it in 15 years of NiMH use. I lost some expensive equipment to alkalines, before I wised up.
I recommend Panasonic eneloop cells and a decent smart charger. If you use high-intensity flashlights, you should get a charger that will also charge the lithium cells for your flashlight. They're readily available