Mac Pro 3,1 Boot fail diagnosis
Hmm, the two week break may have killed the Power Supply Unit (PSU), especially if it was a rather cold morning when you tried to first power it back up. It wouldn't be the first time I've seen that happen. :(

As for using a multimeter to test the PSU, I wouldn't bother since the LEDs indicate the PSU is still providing trickle power and I wouldn't really encourage trying to test around some of the other rails (just to be extra safe). 

However, a fairly simple test you can do is a system reset. To do this:


Power up the Mac Pro
Wait for it to get to its current usual state
Press the SYS_RST switch on the logic board (yep, this needs to be done while it's powered on). 
If the Mac Pro boots, shut it down again
Now try booting up again from the normal power switch
If it doesn't boot, press the SYS_RST switch on the logic board again (don't power off first)


Basically, if your Mac Pro does boot from the SYS_RST switch but not from the main power button, then it's most likely a new PSU you'll need.

However,let's not give up yet! I'm not sure if you have a full Mac Pro Service Manual at your disposal, but before purchasing another PSU it may be worth going through the following process.

Just ignore anything obvious or that you've already tried - I'm just trying to write this so it may be useful to others too. Obviously, before purchasing another PSU, it's worth trying anything below you haven't already tried.

Firstly, let's summarise what to make of the LEDs:


LED 1 off is a Good sign! - You wouldn't expect this to be lit unless your Mac was powered up but in sleep mode
LED 2 on is a Good sign! - This is normal when pressing the DIAG_LED button
LEDs 3 and 4 off is a Good sign! - These LEDs only come on if an error occurs or if the BootROM is corrupted (and you don't actually have to press the DIAG_LED button to see them)
LEDs 5 and 6 off is a Good sign! - Normally off. These LEDs actually have two indicator modes (i.e. they can either stay on or flash, depending on the error). Regardless, not an issue here
LED 7 off is a Bad sign! - This is normally on when pressing the DIAG_LED button. See troubleshooting suggestions below
LED 8 off is a Bad sign! - This is normally on when pressing the DIAG_LED button. See troubleshooting suggestions below
LED 9 on is a Good sign! - This is normal when pressing the DIAG_LED button (assuming the computer has been powered up for about 5 or do seconds first)


Troubleshooting LED 7

LED 7 should be on when pressing the DIAG_LED button.

Typically to troubleshoot this you would:


Check that the graphics card is seated correctly in its PCI slot
Check that the card’s auxiliary booster power cable is connected properly (i.e. if the card requires one)
Try the graphics card in a different PCI slot
Try another graphics card if you happen to have one handy


Troubleshooting LED 8

Likewise, LED 8 should be on when pressing the DIAG_LED button.

To troubleshoot this:


Check the power cable connections
Swap your AC power cable with another one and test again
Unplug all external hardware (except for the display, keyboard and mouse, but also including any UPS you may be connected to) and test again
If you happen to have another USB keyboard you can try, use that instead while booting up 
Check for any signs of an obvious electrical short, e.g. metal screws or a loose PCI card slot cover touching the logic board. Also check the logic board for any soldering that’s not quite right (something loose, or something that could come into contact with something else)


Other troubleshooting

Some other things to try include:


Reset the SMC and test again
Reset the NVRAM and test again
Remove the battery for about 30 seconds, insert it again and test
Although you've only fairly recently replaced the PRAM battery, it's cheap enough to just try replacing it again