At this stage you don't need xboxhdm, or an exploitable game. Your Xbox is booting Frosty's Rescue disc, so it is still modded, and you still don't know it it's softmodded or flashed, although softmodded is more likely.
There are lots of possible next steps, but the easiest is to enable DHCP on your network, then use Frosty's and FTP to replace the files you deleted. You can then FTP BIOS checker across, and run it to see which BIOS your Xbox is running, which will allow you to work out if the Xbox is flashed or softmodded.