iMac 13,2 (late 2012) suddenly very slow. How to make a diagnosis?

Your hard drive is failing, and that’s why your iMac is slow. If you use an external SSD and run macOS from that, it will make your Mac 10 to 30 times faster for storage, AND you no longer would be running from a failing drive. For more info, instructions, and what external SSD to buy, please read: https://discussions.apple.com/docs/DOC-250003583.

Jack

Jack, FYI, the report ended up on a new thread in "Older Hardware." I answered and had it moved here before I saw this one:

VERY SLOW iMac (late 2012) since last week - Apple Community

That Fusion drive is a hot mess.

Welcome to the Apple Support Communities.

I believe we need to proceed with an EtreCheck report. EtreCheck is a useful App to diagnose and identify issues, both hardware and software. It is trusted by many users. Please follow the steps outlined below:

1. Navigate to: https://etrecheck.com and get the free version of EtreCheckPro.
2. Download it. It is a free and trusted app. It is often recommended here on Apple Support Communities. 
3. Open EtreCheck, and before running the report, be sure to check “Allow full Drive Access” at the bottom of the EtreCheckPro window.
4. Run the report, and put it in a post, as seen here: https://discussions.apple.com/docs/DOC-250000211. We will then examine the report for issues in your Mac.

Cheers,

Jack

Sorry, there is no report attached.

OK, thanks! I missed that.