It's basically completely untrue, at least for the two monarchs that I know very well that ruled in 1802. In Prussia you had the somewhat meek and ineffective ruler (compared to Frederick the Great). However he was level headed and generally not insane. The other strong example to go against this quote would be Francis II of the Holy Roman Empire. While autocratic and old fashioned, he was certainly stable minded but not very sociable.

Christian VII was regent of Denmark and Norway between 1766-1808. Throughout his rule, Christian was largely neutralized and the kingdom was de facto ruled by, first, the king's physician, Johann Struensee, then from 1772 the Queen Dowager Juliane Marie, and finally from 1784 until Christian's death, by his son, Frederik. It has been speculated that he suffered from schizophrenia (Viggo Christiansen: Christian den VII's sindssygdom, 1906), but later historians are more reluctant to diagnose. Ulrik Langen (Den afmægtige, 2008) dismisses any diagnose and focusses on other factors, such as the king's young age at his father's death, the state of education and upbringing in the royal family at the time, and the power of the Council of State.

Christian suffering from "mental illness" is still conventional wisdom to this day, and that, combined with his marriage to Caroline Mathilde, sister of King George III, suggests to me that Bagehot probably had Christian in mind as one of his "insane monarchs".

Given that 1 in 4 people are said to suffer from mental illness at some point in their lives, would it be statistically significant if every Royal family had a member with a mental illness in 1802?