A little.
I know you tried Recovery Mode, which attempts to install the last version you had...have you tried Internet Recovery? It will install the version of OS X/macOS that your iMac came with. You can then upgrade to any later versions using App Store > Purchased.
Internet Recovery - power on or restart and hold Command-Option-R immediately when you hear the startup chime. Keep holding until your see the Starting Internet Recovery globe icon. You should then get prompted to join a Wi-Fi network. Once joined, you should see a progress bar, then the OS X Utilities screen, where you can choose Reinstall OS X. It can take quite a while for this to appear, since it is loading from Apple's servers over the Internet.
See how that goes,
C