Cannot connect to Team Foundation Server in Visual Studio 2012 after update 4

I installed Visual Studio 2012 Update 4 (downloaded from MS download center) and lost the connection to Team Foundation Server from Team Explorer. Before that, I was connected right at VS2012 startup. It is not the case anymore and I cannot connect to Team Foundation Server from any entry in VS (Team menu or Team explorer window).
For instance, nothing happens when I click this link:

No modal error window, no indication of an error occuring in windows event viewer or in VS ActivityLog file in roaming directory, so I assume that there was no error.

I tried these and it changed nothing :

• Repair VS2012 from Windows configuration panel
• Install Team Explorer for Microsoft Visual Studio 2012 based on this blog post: Connecting to TFS
• Restore my system from the restoration point of VS2012 update 4. I got this error at restart : An unspecified error occured during system restore : (Visual Studio 2012 Update 4 (KB2707250)). additional data : 0x80070005.
  
• Install VS Update 4 in system admin mode but the result was the same (corrupted VS install)
  

The only working part left of VS was the source control explorer. Appart from starting and using source control explorer, VS is unusable. e.g. when I started a web application using VS development server, I got an error message saying that my VS installation was probably corrupted. The only way around was to uninstall and reinstall VS. Some of my co workers have installed Update 4 and do not experience the same issue. But this Stackoverflow question is about the same bug.
Team explorer window is very different without Update 4.

From event viewer, I see errors which all have the same pattern. Their occuring time match my attempts to install VS 2012 Update 4
Here are their data (I have filtered and removed duplicates)

• .NET Runtime Optimization Service (clr_optimization_v4.0.30319_64) - 1>Failed to compile: C:\Program Files (x86)\Microsoft Visual Studio 11.0\Common7\IDE\TF.exe . Error code = 0x80131f07
  
• .NET Runtime Optimization Service (clr_optimization_v4.0.30319_64) - Failed to compile: C:\Program Files (x86)\Microsoft SDKs\Windows\v7.0A\bin\x64\aspnet_merge.exe . Error code = 0x800700d8
  
• .NET Runtime Optimization Service (clr_optimization_v4.0.30319_64) - 1>Failed to compile: C:\Program Files (x86)\Microsoft SDKs\Windows\v8.0A\bin\NETFX 4.0 Tools\WinMDExp.exe . Error code = 0x80131f07
  
• .NET Runtime Optimization Service (clr_optimization_v4.0.30319_64) - 1>Failed to compile: C:\Program Files (x86)\Microsoft SDKs\Windows\v8.0A\bin\NETFX 4.0 Tools\SecAnnotate.exe . Error code = 0x80131f07
  
• .NET Runtime Optimization Service (clr_optimization_v4.0.30319_32) - Failed to compile: C:\Program Files (x86)\Microsoft Visual Studio 11.0\Blend\Microsoft.Expression.Blend.WwaClientPS.dll because of the following error: The module was expected to contain an assembly manifest. (Exception from HRESULT: 0x80131018).
  
• .NET Runtime Optimization Service (clr_optimization_v4.0.30319_32) - Failed to compile: C:\Program Files (x86)\Microsoft Visual Studio 11.0\Blend\Microsoft.Expression.Blend.WwaClientHost.dll because of the following error: The module was expected to contain an assembly manifest. (Exception from HRESULT: 0x80131018).
  
• .NET Runtime Optimization Service (clr_optimization_v4.0.30319_32) - Failed to compile: C:\Program Files (x86)\Microsoft Visual Studio 11.0\Blend\Microsoft.Expression.Blend.WwaClient.dll because of the following error: The module was expected to contain an assembly manifest. (Exception from HRESULT: 0x80131018).
  

Here is an export of my corrupted VS2012 installation info :
Microsoft Visual Studio Premium 2012 Version 11.0.61030.0 Q11REL Microsoft .NET Framework Version 4.5.51209

Installed Version: Premium

Team Explorer for Visual Studio 2012 04941-004-0043007-02023 Microsoft Team Explorer for Visual Studio 2012

Visual Basic 2012 04941-004-0043007-02023 Microsoft Visual Basic 2012

Visual C# 2012 04941-004-0043007-02023 Microsoft Visual C# 2012

Visual C++ 2012 04941-004-0043007-02023 Microsoft Visual C++ 2012

Visual F# 2012 04941-004-0043007-02023 Microsoft Visual F# 2012

Visual Studio 2012 Code Analysis Spell Checker 04941-004-0043007-02023 Microsoft® Visual Studio® 2012 Code Analysis Spell Checker

Portions of International CorrectSpell™ spelling correction system © 1993 by Lernout & Hauspie Speech Products N.V. All rights reserved.

The American Heritage® Dictionary of the English Language, Third Edition Copyright © 1992 Houghton Mifflin Company. Electronic version licensed from Lernout & Hauspie Speech Products N.V. All rights reserved.

Oracle Developer Tools for Visual Studio 11.2.0.3.20 Oracle Developer Tools for Visual Studio Copyright (c) 2005, 2012

PreEmptive Analytics Visualizer 1.0 Microsoft Visual Studio extension to visualize aggregated summaries from the PreEmptive Analytics product.

SQL Server Data Tools 11.1.20627.00 Microsoft SQL Server Data Tools

You need to click the configure team projects link.