I don't think shutting it down suddenly would cause this, although moving it could definitely break something. If iLO is working, it should register with your DNS server using the server's serial number. For example, if the server's serial number is USEABCDEFG, and your domain name is domain.com, then the host name would be ILOUSEABCDEFG.domain.com. Perhaps you could reach the iLO Web page there. The serial number and iLO credentials are on a toe tag on the front of the server. I would try hooking up the iLO port to a known good switch port and seeing if you get a link light on the iLO port and then try that host name.
Without logs or error messages, it's going to be tough to troubleshoot. It could several things, including a bad power button (it's rare, but this actually did happen to me once), a bad power backplane (these are replaceable on their own), a bad CPU, etc. You mentioned all the cables have been reseated, but have the boards themselves been reseated, as well? The network ports, iLO and the BIOS are on an "SPI board" and that connects to the system board. The power backplane connects to the system board, as well. The CPU and RAM are on their own board in the memory drawer and that connects to the system board. Perhaps one of those connections wiggled out slightly? I know if you call HPE support (800-633-3600), they could send out a CE who would probably just start yanking parts out and putting in new ones until it boots. They'd charge you labor and parts to do it, though, since I'm sure your server is out of warranty by now. They do troubleshoot things for free with you over the phone, though.
