I have a few ideas here.
1) I noticed that you have the Motherboard out of the case in your pictures. I tried the same method of test when I first acquired my 5162. It refused to come to life until I put it back in the case. My theory at the time was that the motherboard needs to be grounded by the mounting screws, but no one else that I've communicated with has duplicated this result, so there might have been something else keeping mine from coming to life when I first tested it. Also, the PSU needs an adequate load before it will start. Some have noticed that the motherboard alone is enough load but others have found that more of a load was required. Did you hear any beep(s) from the speaker when you tested with the other PSU? Was the speaker connected?
>>>> Speaker was connected, I heard no beep after 5 minutes.
2) The keyboard in your picture won't work with the 5162. You need an AT keyboard. Did you try the other keyboard with the switch in the AT mode?
>> Ok, so confirmation that the system requires a AT keyboard, great... I have an IBM model M (best keyboard in the world), I will use it exclusively.
3) You may need to try a different video card. Just because that card worked in a 5150 doesn't mean it'll work in the 5162. If you have a 16 bit VGA card, try that.
>> I have a few 16-bit ISA cards in my collection, I will try them all - including a ATI mach32
Anyway, that looks like a nice example of a 5162. With a little work, you should be able to get it working.
Greg
Edit
Looks like I was to slow with my reply and a couple of others have mentioned some of the same things that I did. I don't mean to be redundant, just slow and lazy.