You don't say why your sister wasn't willed a share of the home, how long she cared for your mom, and which family member -- perhaps you -- is executor or was given durable power of attorney to make decisions. All of this, as well as the nuances of your state's laws (they're all different), could have a bearing on her rights. If your sister claims there was an oral agreement between your mom and her regarding use or possession of the house following her death, this could further complicate things.
Barring any equitable arrangement you can strike with her, within the legal bounds of your mother's stated wishes, you may have to hire a probate or estate lawyer to iron this out.
Before that happens, let's look at a few realities. Though you may harbor animus toward your sister, as your quotations around the word "caretaker" seem to indicate, realize that caring for an ill and elderly parent can be a round-the-clock grind and prevent a caregiver from holding a job and advancing a career or having much of a life at all. If no family member steps forward to provide this valuable service in such a situation, then a full-time, live-in caregiver typically charges between $150 and $250 per day. So, your sister is likely entitled to at least something if she performed this function for a lengthy period. In some cases, such family caregivers are even awarded the house in court! So tread carefully, brother. Sister may just decide to file probate and an expensive, ugly legal entanglement could ensue.
It may be best to simply ask her, civilly, what she really wants. One option could include letting her buy out everyone else's share of the house at a discount, though granted, her present job or credit status may not allow this. Another is to allow her to live in the house rent-free for a specific time (say, a year) to rebuild her career before you sell it, provided she pays taxes, maintenance, utilities and other expenses. In some cases, heirs/siblings simply agree to pay the caregiver a lump sum out of the estate for services rendered.
Of course, all of this is further complicated by your family's wish to pay off your mother's debts ASAP. So, the sooner an accord is reached with her, the better.
This is, by the way, an issue that will only grow in stature in the next few decades as such caregiver arrangements become more common with our aging baby boomers becoming senior boomers.
It's also worth noting that if one or both parents served in active war duty -- and there are tons of Vietnam veteran seniors out there -- they're likely eligible for VA "aid and attendance" pension benefits that provide compensation for family caregivers.
As for your family's predicament, Tom, the more civil and aboveboard you are with Sis, the better this will likely shake out for everyone, regardless of her pecking-order status with the rest of your family. Good luck!