If you’re comfortable pulling th iPad apart, then do so. There are a number of parts inside that are connected by flat cables into ZIF sockets; these sometimes work loose. I’ve also pulled apart iPads (not this particular model) and found a ZIF socket broken after a heavy-handed screen replacement. A loose connection can signal a reboot request by mistake. Also check the battery is plugged in properly. Apple don’t really want people poking around inside their gadgets; it’s not designed to be easy. if you’re not comfortable doing this (see the iFixit repair guides), then get a pro to look at it (or get yourself a new one).
This is fairly common on LCD and plasma screens. It can be caused by a number of different things; the most common is the connection to the edge of the LCD. If you search YouTube you can find videos that show how to fix. It’s fiddly to get the TV apart, you’ll need plenty of room, and discipline in labelling the screws so that you can put them all back in to the same places. Be very careful when repairing; you’ll have to run the TV out of its case with exposed connections. I can’t remember if your model uses a cold-cathode high voltage backlight or not; if it does there could be 1000V or more exposed at the edge of the screen.
This is a hard gadget to get apart. However, removing the three posidrive screws (one under a `warranty void if removed' sticker) allows prying the case apart enough to get at the on-off switch. It's a fairly common 3mm square button switch; I replaced it, and the scales work again.
If the power board is OK the power ON light should appear. If you disconnect the harness from power board to mainboard, are the voltages correct? There's a polyfuse in series with the mains input, it could have gone entirely.
Is the laptop totally dead (can you see the splash screen on booting? Does the `charge' lamp light up when it's on AC?) Possibilities I can think of include: --- temporary blowing of a soft fuse becuase of overcurrent drawn from the USB port --- Damage to the power supply I'd try removing the battery and AC, and leaving it for 24 hours, then trying again as a first step, in case it is just a self-healing fuse. If that doesn't work, you're going to have to pull it apart and look for damage on the mother board -- which is a bit of a pain on most laptops, they're not really designed for serviceability.