You and your wife have every right to have your home back if your mother-in-law's behavior and sister-in-law's interfering are intolerable. No one can force either of you to personally
provide your mother-in-law with housing and care, not even the state. It is necessary, however, that you identify some alternative living arrangements to which your mother-in-law can transition. To do that, you'll first need to find out if she qualifies for Medicaid.
I've discussed Medicaid in other posts so feel free to browse, but let me nutshell-it here: Medicaid will cover the cost of a nursing home and in some rare cases, an assisted living facility. For many seniors, particularly those with dementia, nursing homes and assisted livings are the only options for housing if they cannot live with a family member. Some states offer Medicaid-funded home care but there would still be the issue of which home she would reside in if not yours.
To begin, contact your local Social Security office and find out what the income and asset limits are to receive Medicaid in your state. Keep in mind "“ and this is really important - that if your mother-in-law does not safely fall below the limits, it is still very likely that she can qualify.
If she exceeds the asset limit for example, her money can be "spent down" by paying for the first several months of the nursing home "out-of-pocket" until she reaches the limit. This would make her transition possible right now and, as an added bonus, the nursing home of your choice would love her because the out-of-pocket cost for a bed often exceeds the Medicaid reimbursement.
If she exceeds the monthly income limit she can simply pay the excess to Medicaid in order to qualify. (They'll be happy too.)
Lastly, you mentioned that your sister-in-law controls your mother-in-law's finances, etc. but did not mention if she is in fact Power of Attorney (POA) for your mother. If she is, she needs to understand that role as one of guardian just as a child under the age of 18 would have. If she thwarts your efforts to transition your mother-in-law out of your home, or continues to insist that you have no rights in this situation, I would strongly advise calling Adult Protective Services (APS) in your county. Such a call would bring a professional to the table who could instruct your sister-in-law on her duties as POA and assist in planning your mother-in-law's transition out of your home.