Disclaimer: Mr. Martin owns A song of Ice and Fire. I am not that lucky since I'm not kissed by fire.
Weakness is not an option, not for a Lannister. He has worked hard for this, he has worked hard all his life; performing his duty and forgetting his honour when it was necessary, teaching his children what to do and how to think, bending them to his will to make them wiser and stronger, true Lannisters like himself. A Queen and a knight. It's all he's ever asked for and yet they both have failed him.
Jaime's as good as lost to him forever and Cersei has lost the little power she held over her son, and the kids are useless to him, nothing more than small pieces for him to play with in order to win this whole game, not only the first hand. As long as Jaime is a prisoner to the Stark boy and he is far away from King's Landing fighting this foolish war that his golden twins have brought upon themselves and the whole kingdom, he'll have to trust the only remaining tool he has with a bit of common sense in him. As long as he remains in the battlefield slaying northerners and risking his life for the kingdom he has built with his own blood and his own sweat, he will need someone to hold it together.
He's not about to let it crumble down and become ashes; he'll send Tyrion to the Capital to control Cersei and that vile son of hers until he can return.
It's his kingdom, the era of the lions, and it will remain as such. For Joanna.