Kodak was a film company. An expert in film. They couldn't save themselves because film died.
They could have started selling Kodak toothpaste or digital cameras but that would not have 'saved' the company. It would have just been a different company with a very similar sounding name.
Well, film is died because Kodak acted like Nokia on R&D in the last years. Sigma does a nice work using the film concept (3-layers RGB sensor). Color rendering on Foveon sensors are the best in the market.
Also, even the chemical film could be innovated, making processing and scanning easier to the end user. However, in the last years Kodak concentrated their efforts on making cheap digital cameras and pushing the printing business. It isn't the film that was becoming obsolete, but the idea of printing pictures in paper. An easy develop-and-scan (share) device would probably make film live longer.
By the way, I still take pictures with my 135 reflex cameras sometimes. I have a Nikon Coolscan V (launched in 2005) scanner which outputs 20MP digital pictures in 14-bit per channel. The results are pretty impressive even in 2012. If I had a 120-format camera and a Nikon 9000 scanner, the results would pair most recent medium format digital cameras.
In short, it's not the film that died, but the way Kodak (and Fujifilm, although Fuji is acting pretty well in the digital camera market) carried its business.