What hardware can drive one long row of 100 LEDs

You're right. The spacing would be a little tricky to work out. The spacing won't be even due the the fact that the center of the black keys are right between the white keys. Printing examples on paper is a great technique though. Instead of trying to do the math, I would just keep increasing/decreasing the spacing until it fit. The goal is to have each LED centered on each key. I would do it in octaves though. From c to b. You don't need to use the 3mm LEDs. You can use the really tinny ones if that is cheaper.