Try booting the iMac into Internet Recovery Mode using Command + Option + R to attempt to access the online macOS installer. In theory this should boot into the macOS 10.13 High Sierra online installer as long as this iMac at some point in its past had macOS 10.12+ installed. If you can boot the macOS 10.13 online installer, then you need to launch Disk Utility. Within Disk Utility click "View" and select "Show All Devices" so that the physical drives appear on the left pane of Disk Utility. Select either the internal drive or your new Kingston external drive (make sure to select the physical drive and not just a volume) and erase the physical drive as GUID partition and MacOS Extended (Journaled). The physical drive is usually identified by it make & model, or an external drive may be identified by the USB name of chipset used in the adapter or enclosure.
If you find that you have two internal drives, then one will most likely be a small SSD and the other internal drive will be a large hard drive. If you have both an internal SSD & Hard Drive, then you will want create a Fusion Drive setup instead of erasing the drive. Use this Apple article to create a Fusion Drive:
How to fix a split Fusion Drive - Apple Support
Sometimes booting into Internet Recovery Mode will only boot into an older macOS installer. If you end up booting into a macOS 10.7 to 10.10 installer, then instead of erasing the drive you will need to partition & format the drive using the instructions in this article:
https://www.owcdigital.com/assets/support/support-formatting-and-migration/Mac_Formatting_6-10.pdf
Unfortunately many people have issues reinstalling macOS 10.7 Lion. If you find you are booting into the macOS 10.7 Lion installer and getting an error, then please provide the exact error message.
The other option for reinstalling macOS is by purchasing the original OSX 10.6 Snow Leopard DVD installer which originally shipped with your iMac which is a DVD with a gray label with the iMac's exact model designation printed on the DVD. Just make sure you have a 2010 iMac since any an earlier model iMac would require a different DVD version of OSX 10.6 Snow Leopard (a retail version). If you happen to have a later model iMac, then there is no DVD installer. You can try confirming the exact model of iMac you have by entering the serial number here:
Check Your Service and Support Coverage - Apple Support
Keep in mind that erasing a drive, or partitioning & formatting a drive, or creating a Fusion Drive will destroy all data on that drive or drives.
You can choose to install macOS to an internal or external drive. Keep in mind that the internal hard drive on such an old computer may not be working or working well. There are ways to check the health of the internal drive(s), but it really requires booting into a full version of macOS (either on an internal drive or an external drive). You can try running the Apple Diagnostics to see if any hardware issues are detected, but unfortunately the diagnostics don't detect most types of drive failures. If you can install macOS to an internal or external drive, then I can provide instructions on how to check the health of the internal drive(s).
