I have no idea if this will help anyone else or not, but:

If you’re replacing the battery (or a bunch of stuff) in a Nexus 4, and the phone does the following:
- Refuses to charge from USB/won’t bring up the “charging” screen when powered off and plugged in
- Won’t power on normally, but if you hold the power button for 10 seconds or so, flashes the red notification LED once when you let go
- Powers on normally with the recovery mode power/volume down sequence

Try a different battery.  I found a bad battery in a new batch that behaves like this, and while I have no idea what’s wrong, the phone was perfectly happy with a different battery (and had been happy on the previous battery).

It took me an hour or so before I tried a different battery, so hopefully this saves someone some time!