In the books, it seems that Jon Arryn hasn't done a lot to curtail the power/influence of the Lannisters.
The Small Council is in the best position with Stannis, Renly and Baelish on it, but those three aren't really united in a power block to support Robert.
The kingsguard has been flooded with knights who are more loyal to Cersei/the Lannisters then the crown/king. How come Jon Arryn didn't exert more influence on Robert so he can name knights from others kingdoms, that would be loyal to the crown, not the queen/Lannisters?
The same thing seems to be true for the Gold Clocks, with Janos Slynt being their Lord Commander. How come Jon Arryn didn't push for a knight from the Vale/Stormlands to gain the position and be an ally to himself and Robert? I imagine that as Hand of the King he would have been able to exert a great deal of influence and stack the positions on the Small Council, in the Red Keep and in King's Landing further in his favor and in the kings favor.