The passage actually started earlier than the quote. I am not going to bother looking it up... but I will set the scene before I start restating and explaining.
Prior to the quote, something was explained about, perhaps, a politician's argument. Or an argument made by a group of politicians.
My restating will always be in italics.
This logic was meant to strengthen Howard's aspiring workers,
The argument made by the politician(s) was intended to have some kind of positive effect on the workers.
I'm sorry, I don't know who Howard is, or why the workers are aspiring.
... just as it made a certain sense of the Peabody Trust's housing rules imposed on the poorest of the poor in the nineteenth century
And the argument made by the politician(s) also somehow digested the Peabody Trust's housing rules.
Let me explain this part a bit more, since it gave you some trouble. This is related to the concept "There's method to her madness". The author is saying that the Peabody Trust's housing rules were unethical -- we know this because of the rules being "imposed on the poorest of the poor" -- but, in some strange way, the argument made by the politician(s) helped the Peabody Trust's housing rules appear valid. Perhaps the politicians' argument helped justify the housing rules, or made them more palatable to some group of people (that I might be able to identify if I had read the whole chapter or section).
In other words, "made a certain sense of the Rulese" means helped the Rules make sense in a weird way. Think of one of those joke proofs where you seem to be making a valid logical chain of assertions, and somehow you end up with an absurd conclusion and then you laugh. So, if you want a one-word synonym for certain here, it would be weird.
rigidity begetting independence continues today
We can see the same stupid reasoning today, where wrong-headed people claim that rigidity leads to people being more independent.
This is shorthand for wrong-headed people support a point of view which is actually quite rigid, or which supports rigidity (sorry, I'm not sure which).
as the logic of military-style camps for juvenile offenders in America.
We see this occurring today, as well, when we consider the irony of the government setting up military-style institutions to house teenaged "criminals" in the U.S., with the military-style institutions being exactly opposite to what those teens actually need.
Note, "as the logic" means similar to the logic.