How can one overcome the obsessive whispers of feeling close to death or insanity that plague a person even after repentance and prayer?

Remembering death is legally مطلوب (mandated/recommended) because it hastens repentance and increases contentment and zeal in worship. It is also one of the signs of good deeds. However, it must not lead to despair of Allah's mercy. Excessive fear of death or madness might be among the whispers of Satan. Therefore, one must seek refuge in Allah Almighty, persevere in His obedience, and not surrender to Satan's whispers.