Why do you knead it for 20 minutes?
That is too long and it is starting to dry out. IMO and experience, nothing brings it back after it starts to dry out.
You should only need to knead to 5 minutes at the most to get it where you can roll it out. If it is a large amount for several tiers of cake, do it in sections. Always keep it in a tight, airproof plastic bag. Put each section in the bag after you knead it to keep it moist.
As for rolling it out, we use The Mat at my shop. I teach all my girls to use The Mat. Once I let a girl do it her way and had a dry, cracking mess on the cake. Now I get *itchy and stubborn and make them learn it my way. It is my name that goes out the door, not theirs. So they learn it my way. We have very nice fondant covered cakes with no cracks, drying, elephant skin or pleats - all thanks to The Mat.
And yes, we use Satin Ice.