I suspect that Light's father getting sick is all staged by L.
- Both are sitting there talking and suddenly both of their phones ring.
- The phones ring right when L is telling Light about the task force.
- L notifies Light about his father, thus proving that he is indeed on the task force.
- Light's father blows L's cover and even uses the name "Ryuzaki" in front of Light.
- "Ryuzaki" is only supposed to be used within the task force.
- L does not mind that Light's father just blew his cover.
The only way for this sequence of events to make sense is if the whole thing is staged.