Reinstalling OSX

Not Totally Sure But You Have Two Options

You have two obvious options:

1. Suck it up and just buy ML from the App Store and reinstall.

2. Take it to your local Apple Store and have them reinstall ML. It would be wise to bring your Proof of purchase with you. They may only do this if you have AppleCare though. Be careful.

 

start the Mini using Command R and it should automatically do it for you.

 

1. + 2. Nonsense.

Hold Command-R, this will get you into recovery, from there you can reinstall the OS.
If that is unavailable (Only when the Recovery partition is wiped) you can still use recovery but this is another shortcut and it is called internet recovery, hold Command-Shift-R for this to work, it is build into the firmware of recent Macs +- 2 -3 year old Macs.

OP, why reinstall, you can find out what is wrong without reinstalling, I suggest making a new User Account in Users & Groups Preference Pane, preferably an Admin Account, restart and login into that account and see if the problem is gone, if it is you have problems with your normal Account.

You should also look into RAM, maybe you have too little RAM, when the Mac gets slow, open Activity Monitor and see if there are pageouts, if so more RAM will help.