For a couple of bartop projects that are using a 10" VGA and the width of the cabinets being only 12", I went searching for something to use to light the marquee area needing to be 12" or less.
I found a 6" fluorescent "GE Closet Light" fixture at Walmart for about $5. It is designed to operate off of batteries. I opened it up and soldered a couple of wire leads onto the battery terminals.
I was going to use a 6V wall wart AC/DC adapter but found it operates just fine and just as bright hooked up to a 5V lead from the power supply.
Something to consider....