You have my sympathy.  I, too, am fighting those two aggressive perennials.  The roots do not go down very deep, just 6" or so.  The problem is, they spread out in all directions.  They wrap around each other and every other thing growing in that area.  I find the Snow on the Mountain easier to remove than the lilies. If you do not have any other plants there that you want to save, you can use a non-selective herbicide but you still will have to pull or dig out the dead roots.  I don't know why you need to put down cardboard if you are digging them out.  I think that is meant to smother grass but doubt if it would work on these two as they will just go to the side.  Kill or dig out what you can and then just watch for any trying to regenerate from a stray portion of a root and dig that out before it has a chance to establish.  Good luck.