Why might my limelight experiment have failed and how can I improve it?
Perhaps your flame wasn't hot enough. Real limelights used oxyhydrogen torches, not wimpy natural gas flames.

Or perhaps your limestone chip wasn't just $\ce{CaCO3}$. If it was dolomitic limestone, there was a lot of $\ce{MgCO3}$, too, and $\ce{MgO}$ doesn't do the trick like $\ce{CaO}$ does.