I have an A1312 2010/11 iMac that won't start

If, after Allen's suggestion you still can't boot up, unplug the power cable from the back of the Mac for about 15 seconds. Plug it back in and try booting up. If it starts a boot (even if it can't complete it) that's a positive indication of a failed power supply.

As you've already been informed that model is considered obsolete by Apple and getting it fixed will probably be much more that then unit is worth. And parts may not be easily fond. Apple won't touch it but an Apple Authorized Service Provider will but will be costly.  

Time to start saving up for a replacement.

After leaving if off for 3 months...

Was the computer still attached to wall power or was it unplugged? If the latter, let the computer sit for 24-48 hour continuously on wall power, then try again. It's a long shot but easy enough to try.

When you try to start, do your get anything at all? Chimes? Whirring noises? Anything other than black on the screen? Some iMacs from that era showed graphics hardware failures. They make the proper noises but show nothing on the display.

Before I would put any effort or money into a 2011 iMac I would consider the fact it is considered obsolete and can't run any version of macOS later than High Sierra which is no longer supported.