mac mini stuck on boot
According to this support article, the system requirements call for a Mac Mini from 2009 or newer. The model number A1283 corresponds to each of the three Mac Minis that were manufactured in 2009 (Early 2009, Late 2009, and Server), so your hardware is compatible. The install image has disappeared because you began the installation process; this is supposed to happen, as the image was not designed to be persistent. The reason that the image is not persistent is partly because (if the installation is successful) if your file system were to become corrupted, for example, you would be able to boot from a separate partition that is made at the time of installation, called the Recovery Partition. In your case this did not happen, so you have neither the recovery partition nor the install media. 

To fix your problem you will need to use another Mac, at least to begin with. The first step comes in two, simultaneous parts:


Re-download the Mountain Lion installer from Apple. Afterward, follow the instructions from this article to create an installer on a flash drive. While that's downloading, go ahead and start with #2:
Re-install the OS X version that was originally on your Mini. You'll accomplish this by booting from the installation discs that came with the machine. This is important! Do not use discs that came with another model or generation! Depending on your exact model, this software will be one of the following:

Mac OS X Server 10.6 (10A433; Server)
Mac OS X 10.5.6 (9G2030; early 2009)
Mac OS X 10.6 (10A432; original OS), OR, for the SAME generation, Mac OS X 10.6.2 (10A2234)



This will take a while, so be patient.

The next step is to upgrade your Mini to at least Snow Leopard (10.6.6). Although this really is a requirement only so you have the Mac App store on the machine. This was a requirement to download 10.8, which is available only through the Mac App store. Since you're downloading 10.8 from a different machine, I guess it would be possible to go from 10.6.x to 10.8, but I've never heard of anyone doing it. Probably just the safest to go ahead and get things updated to 10.6.6 or higher.

Once your Mac Mini has been restored and updated, go ahead and run "Verify Disk" through Disk Utility, just to be sure that the file structure is squeaky clean. It can't hurt, even though you've gone straight from a clean installation through some upgrades without anything in between, so go ahead and run it to make sure everything is spec.

Once that's all been done, go ahead and use your flash drive installer to attempt another installation of Mountain Lion. Not all installs work perfectly the first time through, so it may be that this one goes perfectly smoothly for you.

Good luck!