iMac won't "wake up"

If you are using wireless devices to try and wake the iMac, then you may just need to go to: Apple > System Preferences > Bluetooth > select the Advanced... button and check Allow Bluetooth devices to wake this computer.

Apple will not replace the original HD with a SSD! You must have take the computer to either an independent Apple Authorized Service Provider or another typeof shop. I gave up on Sleep years ago and use the Energy Saver - Display Sleep setting. This works and has the same net effect as setting the computer to sleep.

I completely agree with rkaufmann. I gave up on the Sleep function years ago. Ever since the inception of OS X there have been Sleep related issues. And even more lately. I just use Display Sleep and keep it set for 15 minutes so I'm not burning the screen. That way your Mac uses very little power over Sleep and is ready to use at the touch of a key without any grogginess. UNIX based systems were made to be always on and they’ve been running busy servers 24/7 for decades.

Set Computer Sleep to Never and then set Display Sleep to the setting you desire. This will work much much better.