LOTR: What Did Gríma Actually Do to Théoden?

When Théoden, the King of Rohan, first appeared in The Lord of the Rings, he seemed to be in a decrepit state. While he could still somewhat speak coherently in the book, he is portrayed as almost senile in the film. It has not been fully established what exactly happened to him, and J.R.R. Tolkien simply notes in Appendix A that Théoden “fell into a decline under the spells of Saruman, but was healed by Gandalf”. However, the influence of Gríma, Théoden’s chief counselor, on the king is also established in the series.