Kodak hasn't been a film-company for years, and in fact they still lead the industry in many ways. For example, look at the market for high-end medium format digital sensors, Kodak is still at the top of that heap. Of course, the market for $20K digital backs is... pretty small.
Kodak's strategy in the digital age was try to become a technology broker, and let others tackle transforming this tech into actual end product - but it doesn't take a genius to figure out that your licensees and customers will eventually vertically integrate and push you out entirely. Canon built their own sensors, Sony got into the sensors game and pushed Kodak completely out of the consumer sensor realm, and between those two Kodak's doom was all but sealed.
Instead, their consumer-level efforts focused on stupid, low-margin, technologically simple wares like inkjet printers and digital photo frames, all lowest-common-denominator product categories where Kodak didn't have a significant technological edge against its competitors.
Kodak invented the digital camera, but they completely dropped the ball. They needed to ship their own cameras.