It could be one of several things. The key itself (BTW, the batteries in the key do not affect starting the engine, they are only required for the remote functions, such as unlocking the doors), the EIS (this stands for electronic ignition switch, the part the key "plugs" into), the starter, all come to mind as possiblities. I would have the car looked at by a shop which can read out the trouble codes to see if anything is stored. One easy thing to try is to use another key, assuming you have a second one you can try.
This in effect would test the other key for a failure.