Unplug the keyboard cable, reconnect the battery, then plug in the charging cable (it should boot).
If it still does not turn on in this state, remove the SSD, use a SATA to USB cable to plug in a known working HD/SSD to the USB. With the keyboard plugged in, hold the option key and try booting into the known working drive.
Missing screws won't cause grounding issues, unless the big screws for the battery, or the square battery connector are missing, but it will still boot without these.
Check the underside of the board for liquid damage, and clean with 95%+ isopropyl if any is present (take a picture of areas affected before cleaning though, for reference if needed).
I suspect it is either the power button, sometimes working on it's own, or a fault with the SSD (formatting or bad drive). Try the above and hopefully you can narrow it down.