Obvious "not a lawyer" disclaimer. I'm using commonly understood TV law here.
"fruit of the poisonous tree" doctrine in US law, which basically states that evidence found through illegal means is not usable in court.
Slightly pedantically, "fruit of the poisonous tree" refers more to subsequent evidence you find based on the initial piece of illegally obtained evidence (or the illegal action itself that led you to the evidence - that's a semantical argument).
The imagery here is that if a tree (the origin) is poisonous, anything that stems from it (subsequent evidence) is considered to be as poisonous.
In this case, all evidence pertaining to the sting is the fruit, and the cell phone footage (or the breaking and entering, cfr semantical argument) is the tree.
So could it be argued that the sting operation was not connected to Jane's illegal breaking and entering?
When requesting permission for the sting operation, they would've had to supply a reason for doing so. Agents do not get to freely plan stings without any superior's prior approval, and that approval won't be given blindly. If they genuinely believed the cell phone footage to be valid evidence, or simply chose not to lie to their superior, that means that it is already known and documented that the cell phone footage was part of the evidence that led to approving the sting.
For example, there's an exception called the "independent source doctrine".
There's two related but different exceptions here. One is inevitable discovery, the other is an independent source. What you're talking about, i.e. their existing suspicions, is part of an inevitable discovery argument, not an independent source.
For inevitable discovery, you need to be able to prove that they would have decided to do the sting and gotten approval even if they never saw nor mentioned the illegally obtained evidence.
Can it be done? Sure. But there is going to be strict scrutiny on such a claim (as a matter of protecting civil rights), and it's really hard to reasonably argue what you would've done if you hadn't known what you did in fact know but shouldn't have.
A reasonable example here would be if investigators illegally obtained the password to an encrypted drive, but the drive itself was obtained legally. They would've cracked the drive without the password eventually, so they would have inevitably discovered the data that they just happened to find faster by knowing the password.
For an independent source argumentation, the same evidence has to be collected legally in a completely unrelated manner. For example, a scorned lover posts the video online or directly sends it to the police, or a legal house search finds another copy of the video. Again, fruit of the poisonous tree applies - if there is any suspicion that this second source is not independent, i.e. the other house search relied on the illegally obtained evidence, then it's fruit of the poisonous tree. It has to be truly independent.
So again, strict scrutiny will apply and it's going to require someone (not involved in the investigation) to coincidentally come forward with the evidence, or police to coincidentally find the same evidence elsewhere. You can't make it happen, because then it's not independent.