You should do the iTunes test to determine whether the screen itself died. If the iPhone is seen by iTunes, you simply need to replace the screen. You can find the detailed repair guide on iFixit; that guide will have links to the needed tools and parts.It is possible that the screen died, since it's badly damaged.
Step 2 would be to follow iFixit's guide for battery replacement of the iPhone 5. But don't remove the battery, just unhook it, press Power Button for 15 seconds, then reconnect the battery and the screen, and try to start the phone.
Step 3 would be to try the logic board in another known good housing. If it still doesn't start, then the issue lies on the logic board. Which would have to be investigated by a pro tech to try and determine the cause.