2010 iMac will not power up at all.

With a failed PRAM battery (the coin battery you mentioned) your Mac would start up but might present with a blank or black screen. Or boot up with an incorrect date/time that then corrects only when connected to the internet (which corrects the date/time automatically).

As den.thed sugested, you probably have a more major hardware failure. Parts for a Mac this old are not findable except through questionable sources like eBay where you will just be purchasing someone else's 14 year old used and probably problematic hardware. The power supply is a likely candidate for the failed part. I don't think keeping this Mac would be worth the expense. If you cannot afford a new Mac, I would get a used one with a warranty from Apple or from OWC, the best one you can afford.

If you are somehow convinced that you want to try replacing the button PRAM battery, I would take the computer to an Apple Certified Service Provider -- Apple's web site has names and locations of them. It is fairly involved so there would be a charge, maybe $100, and it is likely not the cause based on the symptoms you describe.

That sounds more like a failed power supply or some other serious hardware issue.