If you have removed the power supply, a good hi-res picture of both sides of the board MIGHT reveal a problem. I don't have experience with this particular model of power supply to make better suggestions, though it would appear that you are not alone with a failed power supply in a late model Intel iMac causing the same symptoms. Some complex Switch Mode Power Supplies have the ability to shut-off completely if a faulty condition is reached, only clearing the shut-off when mains power is removed for an extended period of time - NOTE I'm not suggesting this is a permanent solution, but it may assist in troubleshooting. The other thing to try is to unplug the mains power for an hour, and try again. I've seen people adapt a standard ATX power supply as a troubleshooting/permanent solution, BUT, without a pinout guide of the 27' Late 2013 PSU, I'd recommend against it, simply because applying the wrong voltage to the wrong pin could damage your motherboard. It would seem that the second power supply is NOT powering up. When you press the power button, this powers up the other part of the power supply which provides higher current and additional voltages to run the machine. The iMac 'Core i5' 3.4 27-Inch Aluminum (Late 2013/Haswell) features a 22 nm 'Haswell' Quad Core 3.4 GHz Intel 'Core i5' (4670) processor with four independent processor cores on a single chip, a 6 MB shared level 3 cache, 8 GB of 1600 MHz DDR3L SDRAM (PC3-12800) installed in pairs (two 4 GB modules), a 1 TB (7200 RPM) hard drive, and a NVIDIA G. A standby power supply which provides a small +5 volt rail to allow for soft power on.

It contains two power supplies in one board.

However, my first port of call would be the power supply. Sadly, it's near impossible to prove without swapping in a replacement module.

Imac 27 late 2013 logic board