My guess is since you have remamants of failed installation still present in your system that is why you cannot install the new instance. This can be fixed by deleting few registry keys related to SQL server. Follow below method , this method was given by Microsoft forum team and is recommended way to remove have helped me and forum users many time . If you have multiple instances below suggesstion would not work so dont use it.
First please refer to below Microsoft support link for what actions to follow before removing SQL Server instance. Its is for 2008 r2 but would be applicable to 2012 as well.
If still you face problem you might as well try below suggesstion which involves removing registry entries.
Note: Removing necessary registry entries might cause inconsistency so backup registry before doing changes.
•1. Uninstall the existing SQL Server and all the components from the add remove program.
•2. Backup the registry . Please read in details the MS link before proceeding.
•3. Delete the following keys using regedit:
◦--HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Microsoft SQL Server
•4. Go to HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Uninstall and delete all the sub-keys referencing SQL Server.
•5. Go to HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services and delete all the keys referencing SQL Server.
•6. Rename all the SQL Server folders in the computer.
•7. Reboot the machine.
If you dont find this useful or have confusion dont proceed. Open a case with Microsoft and allow them to have a look at your system. with my experience with many failed installation registry keys are left in corrupted state and only solution would be to remove them.But you need expert to fiddle with registry
PS: Above method was suggested in SQL server forum and works well.
Of course if you can please revert it would be helpful