First of all, measure the new battery voltage, just to make sure it's charged.
Someone in a blog mentioned sometimes you can reset the unit by pressing the on and mode buttons simultaneously, for 30 seconds, so you might try that first.
Second thing to check is that the battery terminals, both on the new batteries, and the terminals inside the camera, are clean. Use a qtip and alcohol or a pencil eraser. Terminals and contacts oxidize over time, mainly due to humidity and condensation. Other than the battery contacts, you could be having a switch or any other electrical contact problem inside the unit. Which means, it will have to be disassembled to be fixed. These are precise, compact devices and I wouldn't recommend self repair. Other than lightly bumping it while holding the on switch, I wouldn't try anything beyond that.
You can take it to a local camera repair shop, or contact Nikon for an estimate- here's their site:
http://www.nikonusa.com/Service-And-Support/index.page
Good luck, and hope this helps.