2011 iMac won't turn on/power up

The story so far: I was out of town for a week and unplugged my mid-2011 27" iMac. When I got back, it didn't respond at all to the press of the power button. No, no chime, no fan sound, no hard-drive chatter, no nothing, as if it was still unplugged from the power source.

I DID perform a SMC reset (disconnect all peripherals, unplug power, wait 15 seconds, plug in, wait 5 seconds, in case you're wondering). Actually, I performed four of them, but have since stopped trying, on the assumption that it was a power issue and unplugging constantly wasn't helping, since unplugging got me into this mess in the first place.

I DID check the power sockets, they are working fine.

I DID keep the power button pressed for 10 seconds, released, tried to power up. It's described as power cycle in a macworld article.

I even kinda tried to do a NVRAM reset, though that went as expected, since NVRAM resets require the system to actually start booting.

I cannot check the cable itself, because I have no replacement, though I'm not sure what damage it should've suffered while lying there undisturbed for a week. I've never had issues with this cable so far. I checked everything was plugged in correctly and not loose.

Other: I'm running El Capitan, but I cannot tell you the exact version number. Not the latest update and possibly not the one before that, either.

There do seem to be hardware issues, there was to a storm of non-trend kernel panics a few months ago, though they have mostly stopped a while ago. I suspect with this type of behaviour it's a RAM problem, though neither AHT nor Rember have been able to confirm any errors. In fact, all hardware tests come out clean. The panics started after I upgraded to El Capitan, so I'm not ruling out a software issue.

I've been keeping the iMac plugged in since Sunday.

I did not suffer power cuts/blackouts/power surges. Neither did my machine.

Questions:

Do these Mac models still have PRAM batteries and can this be the root of the issue? Do I need to get it replaced or will it recharge eventually?

Could bad RAM actually cause this type of problem? (I mean, I guess it could, but is it likely?)

Is there some way to force start the machine?

Is there something I haven't thought of so far?

(I'm really broke right now and I really need this computer to last a couple of months longer...)