Can't get SQL Server to start or connect to my local database engine through SSMS

SQL Server has been working fine but since Friday I have not been able to connect to the database engine in SSMS. This coincided with my having to power the laptop off during a Windows update because it appeared to be hanging (I don't know if these two are related).

Since then I have:

• Gone back to a restore point before the failed update
• Removed SQL Server 2008 from my machine
• Installed SQL Server 2012 onto my machine. This failed during the load
• Removed everything that was labelled SQL Server from my machine
• Redone the failed Windows updates
• Installed SQL Server 2012 which again failed on load
• Created a self-signed certificate (http://technet.microsoft.com/en-us/library/cc753127(v=WS.10).aspx)
• Tried to start SQL Server from both Services and SSCM with no luck
• Tried various log on options for SQL Server - under the local system the service will start but I still can't connect to the database engine in SSMS
• Read endless posts and articles

Can anyone help me get SQL Server started on my machine, then connect to the database engine using SSMS?

The failure log for the installation gave the following information:

2013-03-05 12:17:37.34 Server  Microsoft SQL Server 2012 (SP1) - 11.0.3128.0 (X64)  Dec 28   2012 20:23:12  Copyright (c) Microsoft Corporation  Enterprise Edition (64-bit) on Windows NT 6.1 <X64> (Build 7601: Service Pack 1)  
2013-03-05 12:17:37.34 Server  (c) Microsoft Corporation.  
2013-03-05 12:17:37.34 Server  All rights reserved.  
2013-03-05 12:17:37.34 Server  Server process ID is 5116.  
2013-03-05 12:17:37.34 Server  System Manufacturer: 'SAMSUNG ELECTRONICS CO., LTD.', System Model: 'RV411/RV511/E3511/S3511/RV711'.  
2013-03-05 12:17:37.34 Server  Authentication mode is WINDOWS-ONLY.
2013-0