These little all-in-one PSU chips are highly stressed parts and not terribly reliable. It's a quart squeezed into a pint pot that is ready to burst out without much provocation. A more substantial power supply would have a separate power transistor and control IC.
The switch remaining illuminated means that the input and output leads on the switch are the wrong way round, so that the indicator is always fed instead of being controlled by the switch. I did wonder why there were bits of insulating tape stuck on the inlet which is not normal. I can't see clearly enough to know how it is wired but the line from the inlet seems to visit the fuseholder first and I'm not convinced the switch isn't in the neutral alone which may be incorrect, but that won't blow up the PSU.
You don't need 230V kicking about to prove that the power would get to where it needs to. I would check for continuity with the power off as it's safer. From the line and neutral pins of the plug to the red and black leads at the screw terminals on the board, should each be a fraction of an ohm. If there's no continuity, check in stages from inlet to switch to fuse etc. just follow the conductor along.
Re the connector, JST, Molex, TE, sorry I can't say for certain. If it were on the bench in front of my I'd go through my parts bins or the RS catalogue looking for a match. You would probably have to buy 100 contacts at 20p each so it might be easier to get a lead sent with the board.