I just tried this and it works. Maybe someone that knows what their doing will give you the proper instructions, but until then.
Plug vga cable into LED monitor. DO NOT plug vga cable into mini. Boot mini. Open system preferences, open displays. Plug vga cable into mini, turn on LED monitor. Choose proper source input in LED monitor's control panel,(my LCD is analog input).
In mini's OSX system preferences display's menu, choose arrangement tab. In arrangement tab you should see two rectangles. If you see only one rectangle, these instructions won't work. If you see two rectangles touching then it will work. In arrangement window, in the smaller rectangle, you'll notice a white band running across the top of the rectangle. Use your mouse or trackpad to drag the white band over to the larger rectangle. Then, you should drag your preferences "DISPLAY" window from your mini's desktop screen to your LED monitors OSX desktop's screen. Choose an appropriate resolution for your LED and that's it. From then on it will boot attached with the vga cable. It starts booting on the mini, but desktop will be on LED. The mini will become the extended desktop. It's not a proper mirror desktop, but if you use search on this forum, you'll find different ways to mirror, other topics address clamshelling, turning off the mini's display, etc. I don't have instructions for these other subjects, but those subjects are answered here, just search, good luck.
I know these instructions have the appearance of being convoluted, but I'm making excuses for being unable to organize, simplify and articulate these instructions, but they worked for me.