You should also begin with a definition of madness. Whatever you are using as your definition will define the criteria you use to evaluate King Lear's behaviour.
For me, madness is synonymous with loss of identity. And though it is by his own instigation, Lear has lost his identity as King when he, in effect, retires. He no longer has the unquestioned obedience of those around him. If he isn't King, who is he? No one is saying, "Yes sir, right away," to his every whim any longer; his daughters are not going to great lengths to provide every comfort to Lear and his many, many knights; in short, he is no longer obeyed without question. He has lost control of his kingdom and the privilege, as King, to behave however he wishes without consequence.
This loss of position and identity as a springboard into "madness" can be seen when he is at his most extreme in Act Four on the heath. Shakespeare uses prose here for Lear's speech -- a signpost in his plays that a high born character is speaking "crazy-talk." Hamlet, for example, chooses prose for his conversations after he decides to feign madness.
However, in IV, vi, Lear does have an extended speech in which he reverts to speaking in verse. Lines 109 - 130 are all lines in which he not only reverts to verse, but to giving kingly orders to imagined subjects (or maybe to Gloucester and Edgar). This grasping at the familiarity of his former power is prompted by Gloucester's (now blind) questioning: "Is't not the king?"
So, the madness of King Lear might be a direct result of his inability to cope with being stripped of his familiar, comfortable role of sovereign. When one is used to be treated and obeyed as the royal "We," it might be nearly impossible to consider who one is when reduced to the simple, fallible "I."
This is an interesting question.
First, ask the question how old is King Lear? He tells us that he that he is in his 80s. When we see him at the beginning of the play, he could be thought child like in the fact that he needs to hear how much his daughters love him. This would appear to be an insecurity on his part or perhaps a true lack of understanding what love is. He seems to think that we are born with so much love and must divide it. To Lear it is reductive not expansive which is the true and mature nature of love.
Senility is often thought of as second childhood. He behaves like a petulant child when he does not get the answer he craves from his youngest and dearest, Cordelia. He throws a tantrum.
Perhaps what we have with Lear is Shakespeare's observation of what we today call Alzheimer's.
He fears going mad. In Act I, scene 4, he tells the Fool, " O, let me not be mad, not mad, sweet heaven!/Keep me in temper. I would not be mad." It is as if he can feel the loss of his senses coming on and is at a loss at how to prevent it.
Lear begins to rely more and more on the "wisdom" of the Fool until he meets the "madman" Poor Tom whose madness is his disguise and survival.
Of course the best example of what is happening to the king mentally is Act III, scene 2 when he challenges the storm. One needs only look into the speeches themselves to see the king's mental state. His speeches are full of punctuation. His mind is in turmoil. The words and sounds of the words also reflect his mental state. The physical storm is reflecting this turmoil, this madness.
He discovers his humanity while mad when he realizes that man stripped of the trappings of status are all just poor naked beings who are the playthings of the gods.
He seems to go in and out of madness after the storm. He has moments of clarity with Gloucester and Cordelia.
So, in the play, you have an old man who might be a victim of Alzheimer's, or mad, or both. You have Edger who feigns madness to stay alive . You have the Fool who may be mad or just crazy like a fox.
As for other examples of madness used to achieve an objective, you could also check out Hamlet and Titus Andronicus.