Can Jane sue for emotional distress?
Anyone who pays the filing fee and delivers a complaint in the proper form to a court clerk can sue for anything.
How likely is she to succeed?
Not very likely at all.
The tort of negligent infliction of emotional distress is not available in this context since there is no relevant physical injury to which it is related (neither her own or someone else's).
The only U.S. state where this theory might succeed under this circumstances is Hawaii and even there, this case would be a stretch because the extreme degree of the emotional distress caused in this case is arguably not foreseeable even if some emotional distress can be expected from all manner of wrongful actions.
So, she is limited to the tort of intentional infliction of emotional distress (also called the tort of "outrageous conduct"), which is a calculated and intentional act specifically engineered to cause her in particular emotional harm, rather than incident to some other purpose (lawful or not) (i.e. basically a mean prank).
It is exceedingly unlikely that this is why this happened (and the title to the question implies that it was merely an error on the landlord or the landlord's agent's part). Even if this was a calculated and intentional effort to inflict emotional distress upon her, proving that this is why it happened would be exceedingly hard in most cases (absent some sort of whistleblowing informant).
If she had a lease with a two-sided attorney fees clause and had to hire a lawyer to establish that she paid the rent and can't be evicted, however, she might be able to recover nominal damages of $1 and her attorney fees fighting the acts taken towards a wrongful eviction as a breach of contract.
But emotional distress damages are not available in connection with breaches of contract with only rare and narrow exceptions not applicable here (e.g. bad faith breaches of insurance contracts, contracts to dispose of a deceased person's remains, and perhaps surrogacy contracts).