Departing employees need to do the less interesting tasks.
But it seems departing employees don't actually need to do the less interesting tasks (and the tasks are self-evidently unimportant because companies don't leave truly important tasks sitting around for a long time waiting for someone to quit so they can be done by the least motivated person in the building) because, accepting your constraints as givens and within the bounds of legal and moral behavior, you have no actual ability to require they be done. If the employee prefers not to do the tasks, and you have no mechanism to provide an incentive to do the tasks, then they're not going to do the tasks.
So first, is she willing to do her regular tasks? The usual non-boring ones that you don't keep around as punishment for departing employees? If she's willing to contribute something of value to the company, then have her do those tasks, even if they're not the ones you want her to do. At least the company benefits that way.
Other excellent answers have suggested gardening leave or paying her to come to work and be idle—not ideal solutions but your constraints have foreclosed most of the other options. But there is one other option: provide an incentive to do the tasks. Right now, there's no incentive because the employee gets paid for the next two months no matter what. If you want to change that, provide an incentive.
Unless there's something else of value that she wants and you can provide, that incentive is probably going to need to be financial. Pay her more contingent on the tasks getting done. You'll want to work out a structure that is fair to both sides in this low-trust environment, so that she believes she'll be paid as promised if she does the work and that you won't be out much if you decide to terminate the arrangement because the work still doesn't get done or it's of poor quality.
It may well seem absurd to resort to paying an employee more to do something you already consider to be their job, but again, you've eliminated the other options already. And if the work is actually truly important to your company, then you'll have to pay someone if you want it done. Compared to hiring contractors, she already knows the office, understands the tasks, and may well be cheaper (because she's still getting her wages). And if you can't trust her to do the work well if you pay her more for it, then it seems you can't trust her to do the work at all, so it's futile anyway.