Chosen Solution
After installing the SSD and replacing the broken HDD, I reassembled the iMac and was able to boot to internet recovery without a problem. Everything worked fine until I got to disk utility, where no hard drives were listed. They weren’t just unformatted, nothing was listed in the side bar. Also, in the install macOS screen no volumes were available to select for be installation. My guess is the power to the drives isn’t connected correctly, but I figured I’d hop on here and get a second opinion before I go taking the computer apart again. Any help would be greatly appreciated :-)
I’m suspecting you installed SATA III (6.0 Gb/s) drives. Your systems SATA ports are only SATA II (3.0 Gb/s). Please list out what you bought for drives and how you installed the SSD drive. Did you follow this guide: Installing iMac Intel 27" EMC 2390 Dual Hard Drive Kit or did you do it this way? Installing iMac Intel 27" EMC 2309 and 2374 Dual Hard Drive If you did the first method then your power splitter might not be good or not fully seated. If you followed the second guide then it’s likely your drives.