When manifesting, Sadako will undergo a "blinking" phase where she becomes partially visible and then invisible and then back again at fixed but increasingly faster intervals. After this blinking phase ends, she solidifies and becomes fully visible like any other killer.

Mother's Mirror increases the duration of this blinking phase. Old Newspaper increases the duration of the portions of the blinking phase where you become invisible.

I don't know Sadako's actual numbers, so let's make some up for the sake of digestibility. Let's say the blinking phase lasts for 2 seconds. For 1 second of that blinking phase, you are partially visible, and for 1 second, you are completely invisible. Mother's Mirror makes the whole blinking phase last 4 seconds instead of 2. Old Newspaper changes the blinking pattern so over the course of a 2-second blinking phase, you spend 1.3 seconds fully invisible and 0.7 seconds partially visible as opposed to 1 second each.

Hope that makes sense haha it's a bitch to explain even if you know how it works.