I was gifted a 2009 iMac.

The following will require any USB keyboard and mouse. If you don’t have them, borrow or buy them, they are inexpensive!

1.     Connect the USB keyboard and mouse to the computer.

2.     Open System Preferences – Bluetooth

3.     Un-pair the wireless keyboard and mouse.

4.     Turn off Bluetooth

5.     Restart the computer in Safe Mode per the directions in How to use safe mode on your Mac - Apple Support

6.     Restart the computer normally.

7.     Open System Preferences – Bluetooth

8.     Turn on Bluetooth

9.     Re-Pair the wireless keyboard and mouse.

10.  Unplug the USB keyboard and mouse and begin testing the wireless devices again. 

You will need to borrow/use a wired keyboard to start the Mac and then be able to pair the keyboard with the Mac.

Well remember a 2009 computer is not much of a gift, it extremely obsolete, has zero value and was likely given to you for those reasons. By obsolete, the last version of Mac OS it can run is at least 5-6 generations old. Parts are no longer available for it and it is slow slow slow. IMHO I'd sell it for parts or recycle it using Apple's free recycle service, Apple Recycling Programs.

Then if you really want a Mac, consider getting a more up-to-date Mac. Never buy from E-Bay, Craigs List, MarketPlace etc..., only buy from Apple directly for either a new Mac or an Apple refurbished Mac. If new or Apple refurbished is out of the question then contact a reputable reseller such as www.macsales.com and consider a used/refurbished Mac.