This is exactly what anchors do! Like, literally! To be 10% of the height, set Y minimum to some value, and Y maximum to that value plus 0.1. To be 20% of the width, set X minimum to some value, and X maximum to that value plus 0.2. Also generally, clear out the “pixel offsets” to all 0 when doing this – the editor defaults to keeping the widget exactly where it was, rather than snap to the new settings you enter.
If you want the aspect ratio of the button to always be, say, 5 to 1, then you’ll need to size the button within the outer container, and accept that, if your screen is more-wider or more-taller than the layout assumed, your button will fill less percentage of the screen.