The odd thing about John is that when he called out the fyrd to support him against the French and the barons supporting him, they went. Despite not having a history of responding to Norman kings. The other king with a bad reputation, William Rufus, also had the support of the common people. And there are some nice legal cases in the Pipe Rolls. John used to go round and take over from his judges (not appreciated - he just couldn't delegate). He pardoned a small boy accused of murder because a stone he threw had killed another, and a simpleton who had been induced to confess to a crime he could not have committed.
John was chronicled by a Roger of Wendover, a man before his time, as there was no Murdoch or National Enquirer to employ him, and his tales were elaborated by Matthew Paris. John was written up particularly badly in the 19th century.
He did have a reputation for womanising, but so did many others, and Philip of France, along with brother Richard, had quite a different reputation.
And although Kipling wrote in a wide variety of styles, it doesn't look like him, the language looks more recent, and he would get his history correct.