that's what our guys used to run wiring through the walls. They weren't running alongside existing wires though, not sure if that will cause problems.

They used electrical tape to tie the cable to the rod and then just pulled it through. 60' is a long way though. You may have to tie a string to one cord, pull it out, tie the HDMI on, and then pull the string to get it back through. Then just pray that it doesn't snag anywhere.