Press hard on Power On button

Not good ;-{

If the button its self is failing its part of the rear housing and not a normally replaceable part (I know! Who won't think of doing that).

But before we jump that far lets see if its just a matter of junk getting in causing it to not work.

First turn off your system and unplug it. Then turn the system around so you have access to the back of it. Now using a cotton swab (Q-Tip) which is slightly dampened with your normal kitchen countertop cleaner (i.e. Fantastic) wipe the seam point around the button and the case. You want to depress the button so you can get the cleaner into the gap but you don't want to flood it! Carefully wipe up the cleaner so it doesn't get too deep. Once you've done this a few times exercise the button is it moving freely? Now wipe down the area with a dry cloth to remove any remaining cleaner and then go over the button again this time with some 85% or better isopropyl alcohol using the cotton swab let this seep into the button gap and then blast the gap with a few hits of can'ed air.

Let the system dry for the night, then plug things back in and give it a try any better?

If not, you'll want to visit an Apple Store or Apple authorized service center and ask them for an estimate of the bench fee + part to replace the rear housing. That will be the cheapest you can get to have someone do it. Otherwise, it's a big job to strip your system down to replace it your self. Here's one source for the part: PowerBook Medic - 21.5" iMac 2011 Rear Housing - Apple P/N 922-9794.

To give you an idea how much work it is take a look at the IFIXIT teardown here: iMac Intel 21.5" EMC 2428 Teardown