What should I do about urine leaking from my infant's diaper onto the carpets and furnishings at home, how can I purify the place, and is it permissible to pray there?

The default principle is that impurity is not transferred and purity remains. Therefore, whenever doubt arises concerning the impurity of something, we adhere to the original state, which is its purity. The certainty of remaining pure is not removed by mere doubt.

As for places that have been soiled by the urine of a child who is over two years old or has eaten food, his urine is like that of an adult. To purify it, it suffices to pour water over the impure spot only, until it is completely covered. There is no need to remove or wring out the furnishings.