Finally, it hit me too: I already heard from three friends that their Nexus 5 died because of the well known "red LED of death" hardware-bug. All three of them were able to exchange their phones within the warranty of the device. But for me, unfortunately, the warranty is over and there is no possibility to get new hardware from LG.
In various forums, there are various proposals to solve this problem, but in the end, most of them conclude with: Buy a new phone.
I didn't want to give up that fast, so I digged deeper and ended up at Youtube videos showing people waving with a heat gun towards a certain chip on the mainboard of the Nexus 5.
Ok, this reminds me of the (somewhat) successful revival of a Playstation 3 in the kitchen oven, so why not giving it a try?
So after disassembly of the device, I put the whole mainboard into the oven. I started with a conservative 70 °C for about 5 minutes, which did not have any effect, the phone did not show any reaction on pushing the power button. Next try was 120 °C for 5 minutes, which brought the Nexus back to life! Drawback: For only about 24 hours. When I powered it down to put the SIM card back in, it didn't want to switch on again.
One more (desperate) try with 180 °C for 10 minutes showed even more success: I am using it since the treatment (about 24 hours), let's see how long it will make it this time.
It will probably go to the boot loop again within the next few days, but this method might be a last resort for people having important data to back up before putting the device finally into the trash can.
(Disclaimer: Sorry, I definitely do not take any responsibility on how your device will handle such a treatment. I guess there is also a good chance that it will be even more broken than before.)