Your going to need a external blank (preferably Firewire 800) drive if your not capable of replacing your Mini's internal drive.
Your going to need a external DVD drive, or use a friends Mac with one and follow these instructions to place 10.6 on a bootable USB thumb drive to install on the external or new internal drive.
To boot from the external drive, hold the option key down on a wired keyboard and select it to boot from, set it as the Startup disk in System Preferences.
As it turns out, I'm currently running Lion, sorry for the confusion. With Lion I can access the Recovery HD and I have a number of options such as reinstalling Lion and using the disk utility (I can even access my standard HD's files which is quite unusual as it won't boot). Is there a way I can create a Lion bootable USB drive from the Recovery HD by redownloading Lion?
You can create a Lion Recovery USB from a friends computer, which is then used to download Lion from Apple's servers to install on a drive.
If you can't do that, then your back to installing 10.6 first, the upgrading to 10.6.8, then upgrading to 10.7, then making the USB above if you ever need it.