Wireless & rechargeable Apple Mouse & Keyboard models compatible with Late 2013 21.5-inch iMac

Yes, but discontinued (look on eBay)

• A1644) Magic Keyboard, first available in October 2015
• A1843 Magic Keyboard with Numeric Keypad

Likewise for Mice. Although not discontinued, the current mouse model differs considerably from the one sold in 2015, specifically in respect of compatibility (current model does not list the 2013 iMac). So search for one on eBay whose description specifically refers to the earlier-sold item ones. Or skip: The Apple Magic Mouse is widely considered an oddity because of its awkwardly-located lightning port (which makes it impossible to charge and use simultaneously).

That information is readily available by looking at each devices (mouse & keyboard's) tech specs. You can find this information in the on-line store. However the newest keyboards also contain Touch ID, your 2013 is WAY too old to support Touch ID.

Please go to Apple.com and click the Accessories tab on the top. Then select Mac and begin your shopping experience. Another option to find accessories that will match your 2013 iMac is Macsales.com, they sell numerous keyboards and mice and offer terrific support and have a number of choices that will work with your 2013 iMac.

Great! Yes, the keyboard too will, I’m willing to bet.

Not the basic 2 port version. Posting for completeness only, glad you’re all set 👍