Leylandii tree stump removal

About 3 years ago I had several Leylandii trees (20+ft high, 30+cm trunks) cut down to ground level. The man cut cross marks into the top surface. Since then the bark's rotted, but the trunks still seem solid.
That section of the garden's since become host to some very happy brambles and other weeds, that are glad there's no big dark trees blocking the light any more, and I've been shamed into doing something to clear it up.
How do I get them to rot down please? I'm not a gardener and experienced enough just to keep the lawn cut.
I don't think I can afford to bring someone in with a digger. I've read in some places that drilling's necessary. But how far in do you drill, what diameter drill bit and do you put something in the holes? I've no idea how deep a tree trunk/root goes into the soil.
Would these trees have harmed my house footings? Some of them were probably only about 10ft away from a gable end wall and they were spaced 4ft apart.