Open Notepad (Start>Programs>Accessories) and open the file "LimeWire.props" in C:\Program Files\LimeWire 1.6d. Your program folder may be called something else if you have a different version or custom named the folder during installation. Towards the middle of the document, you'll find a string that says:

SHOW_TRAY_DIALOG=false

Change the string to this:

SHOW_TRAY_DIALOG=true

Make sure LimeWire is not running while you do this, or it might not work. After saving LimeWire.props and closing Notepad, try launching LimeWire again. Good Luck! :-)