"You can fix something that's broken, but you can't fix something that was missing from the beginning"
The joke however is that this novel is (in principle) an inverted mystery and the reader knows what caused the death of Mrs. Stratton. And it is definitely not what Sheringham is thinking. Jumping Jenny is actually a double inverted mystery that manages to play wonderfully with the format: we first see the events that lead up to Mrs. Stratton in classic inverted style. In a normal mystery, we would see a detective arrive at the scene and have him solve the case. The fun we usually derive from such inverted mysteries, like Columbo and Furuhata Ninzaburou, is usually one of two types: usually you will be cheering for the detective, and enjoyment is derived from seeing the detective slowly, but surely closing in on his target. Sometimes you root for the murderer, because the victim was someone who really had it coming to him/her. For both types, the intellectual battles between murderer and detective are usually the highlight of such stories.
The way I read Jumping Jenny however, didn't fit any of these types. I wasn't rooting for the murderer, nor for the detective. I was hoping the detective would fail. Not because I thought the murderer should have gone free: but Roger Sheringham is portrayed as a character you want to see fail. He is the self-concious Amateur Detective: he comes up with grand theories and notices small things no other people would notice. He is the Thinking Machine of the story. It however also places him in a state of mind other detectives occasionally seem to visit too: he thinks he is always right, and that he has the right to judge. In Jumping Jenny, Sheringham is a) convinced that Mr. Stratton is the murderer and b) convinced that he should try to help him, even if it means perjury and having to fix fake evidence.
So here we have the initial murder, told in an inverted style and then an on-going inverted story where Sheringham is commiting the crime of faking evidence and inputting witnesses with fake memories! And it is told in such a way, that the Detective is the Criminal. In the end, I ended up rooting for the Proper Authorities, which is something you don't often do in novels with amateur detectives.
Considering Berkeley wrote early inverted mystery novel Malice Afterthought (as Francis Iles), the way this novel plays with the inverted mystery is wonderfully meta-concious. In a way, you might consider this an anti-mystery, or at least a critique (and loving parody at the same time) of the flawless amateur detective who can act freely from the proper authorities. It works here great (at the expense of Sheringham), and it makes Jumping Jenny a recommended read.
And somewhat off-topic, but I do think that is kinda ridiculous that the Japanese translated version of this novel is actually a lot cheaper (even though translations are relatively expensive) than the current English version in print.