That Sweet Little Old Lady

Malone lives in a world where psionic powers such as telepathy and teleportation exist. He must cope with them as well as an FBI Director who leaves Malone continually confused about what situation he is being asked to handle and what he is expected to do about it. That Sweet Little Old Lady finds Malone charged with investigating leaks in a secret government program. For assistance, he recruits a powerful psi from a mental institution who believes she is Queen Elizabeth I of England.
The problem is, she may be right. Perfectly marrying SF adventure with slapstick comedy for hilarious results, it was nominated for a Hugo Award in 1960.