Well phooey. My Nexus 5 appears to have suffered the same hardware failure my S2 suffered a couple years ago. The phone doesn't do anything when I push the power button, although sometimes if I push it hard enough it will come on. Plugging the phone in causes it to power up, flash the Google logo on the screen, and then it immediately shuts off and begins the cycle all over again. It doesn't even stay on long enough to get into recovery mode. That also means I can't back up or copy anything off the phone.
None of the usual incantations have helped, so it looks like a call to the Google help center is in store. It's a couple months out of warranty now, so I doubt there will be anything they can do aside from have me send it back to LG and pay for them to repair the phone.
Another alternative is to see if any of the local cell phone fix-it places around here can do that kind of repair (or just replace the board the switch is on) or to find a switch or mother board and replace it myself.
Off to deal with people on the phone.
Update: Decided to ship it back to LG for repair. After speaking with a Google tech support person and doing a bit of troubleshooting with him, he established that it was out of Google's warranty coverage period and put me in touch with LG support. At LG, the first person I spoke to said their warranty coverage was 15 months from purchase, and the repair would probably be covered under warranty. Had to dig around my email archive to find the proof of purchase they needed (it had to have the IMEI number), and when I called back the second person said the warranty coverage was 15 months from the manufacture date. So now I have no idea if the repair is going to be covered under the manufacturer warranty or not, but they have the info they need so I'll let them sort it out there. I'll have the repair done regardless of whether it's covered or not.