In the textbook "Borjas, G.J. Labor economics." (Boston, Mass.; London: Irwin/McGraw-Hill, 2009) fifth edition [ISBN 9780070172708]. Below is a screenshot of the example of payroll tax of 1 dollar on the employer. And the labor demand curve shifts down by 1. What happens if the tax is not 1 dollar but, say, 10% of wage? I know that the labor demand curve will shift downward. But does it also rotates in a way that it has a steeper slope?
Your guess that somewhat is happening with the slope was a good one. The slope will indeed change. If you want to draw that new Demand line ($D_1$) it has to start at the exact same point of the old demand line (if wages are zero the percantage tax will be zero too). The correct intersection with the Y-Axis would be: $(1-t) W_0$ (this simply means one minus the tax rate ($t$) times the old intersection point ($W_0$). I also made a little graph: