The Error message sounds like it is coming from The OS. Typically, IE downoiads files to a temporary folder that disappears the next time windows opens. If you installed LR into this Directory, the the Shortcut referring to LR is pointing to the Temp directory which is long gone.
Right click on the Shortcut icon and when the context menu opens, look at the file location listed in the Target Field on the Shortcut tab of the Properties. Please report back here with the contents of the Target field I just described.
Alternately, Open LR from the Run command by entering it's location directly into the Run Command Field. On my machine, LR is installed in "C:\Program Files\Adobe\Adobe Photoshop Lightroom 2.6\lightroom.exe"
If none of these suggestions result in the app opening, check to see if there is a file named and located in C:\Users\<YourUserName>\AppData\Roaming\Adobe\Lightroom\Preferences\Lightroom 2 Preferences.agprefs
If it is there, delete it and try starting LR.