Have you previously installed SQL 2012? Or did you inherit this machine and don't know its history?
Open the SQL Server Configuration Manager, and see if your instance in listed there. If it's not there, then SQL doesn't think it's there, and vice versa.
Next, look for a file like this:
C:\Program Files\Microsoft SQL Server\110\Setup Bootstrap\Log\Summary.txt
Take a look through the file and see if it sheds a clue as to the problem.
You should also see a series of folders for each of the install attempts.
Also, I would try running the setup program and try the repair option.
Another possibility, is to install a new instance with a new name.
For uninstalling, try downloading CCleaner and running the uninstall through that.