It is imperative that you cease this infernal flirtation, lest you unleash forces the likes of and with which the world has neither known nor been equipped to contend.
First, you have a parallelism problem. When you are applying a parallel structure, which is what you're doing, it all has to still work even if you remove parallel parts. While "lest you unleash forces with which the world has not been equipped to contend" makes sense, "lest you unleash forces with which the world has not known" doesn't.
Now, if the last bit weren't a "neither/nor" but an "and," you maybe could get away with writing "the likes of (which) and with which" in the preceding bit, but only if you the alternatives joined by "and" in the second bit were followed by a comma and the word "respectively"respectively" in order to cue to the reader that the former refers only to the previous former and the latter refers only to the previous latter and that both don't refer to both of the previous."
That said, I don't see any way that you can restructure your sentence so that the latter part is an affirmative statement, thus allowing you to use "and" instead of "or" or "nor."
The way to write what you wrote is:
It is imperative that you cease this infernal flirtation, lest you unleash forces the likes of which the world has not known and with which the world has not been equipped to contend.
After wading through your original sentence to work out what it is you mean, which no reader should have to do, it becomes clear that "the likes of (which)" pertains only to "the world has (not) known" and "with which" pertains only to "the world has...(not) been equipped." Each does not apply to and does not grammatically work with the other, which is required for you to use the parallel structure you are attempting to use.