First, sit down with you Mom - and discuss what her intentions are. Tell her that having a Power of Attorney you want to make sure you and she are on the same page. Second, having Power of Attorney - if the powers INCLUDE the power to sell real estate, yes you probably do. However, it wouldn't hurt for you to sit down with an Elder Law attorney in Florida - preferably one who is a member of NAELA. See Avvo.Com under Find-A-Lawyer. Good Luck on this. I applaud you for wanting to help you Mom and to do it the right way.
You are in a tough spot. While it is possible that you would have the right to sell, you need to be really careful that this would not be considered to be self-dealing. (You are considering selling an asset that you would never receive anything from, while retaining assets that you WOULD receive a portion of). I would try to make sure that all parties are on the same page about what is going on. That could be very difficult under the circumstances. The other possibility that you should review with legal counsel is that the property in MI may be deemed to be "unavailable" to your mother if it is jointly held and your brother refuses to allow the sale. It is not clear from your summary exactly how title it held.
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You should meet with an elder law attorney that does Medicaid planning as soon as possible. The FL homestead is an exempt asset for Medicaid purposes and the MI property may be considered exempt as well since it is jointly owned and the other owner is not willing to sell it.