I'm going to start with your second question first. Your HOR is Michigan. That is what your residency was when you entered into the service, and unless you change your SLR through an approved Form DD 2058, it continues to be your tax home. (You might have had an opportunity to switch your SLR to Florida previously while you had a home, driver's license, etc. in Florida, but that will be a harder sell now. Had you been able to take advantage of this, your wife would also be able to be a Florida resident and not pay state taxes at all).
Because your HOR is Michigan, at the very least your state of residence is not Indiana. Your wife's state of residence shouldn't be Indiana either. It either definitely is Michigan or you can elect for it to be Michigan, which will simplify your taxes going forward. (It definitely is Michigan if she shared Michigan residency with you at any point in time, no matter what state she's from).
If that election is made, then you and your wife should file an Indiana nonresident return. (Because of the military spouse situation, you wouldn't file a nonresident reciprocal return even though Indiana and Michigan have a reciprocal agreement. That agreement doesn't matter when the Military Spouse provision is used; the income is not taxed in the state where you are stationed, period).
Filling out the Indiana return this way will certainly get back all of your withholdings. But the extra payment could be a little tricky. I suggest one of two options. Option 1 is to go to the Estimated Payment section of the tax return and try to include the payment in that section of the program as an additional payment. There probably is not an entry that perfectly describes your mistaken payment, but entering the payment on this screen will carry the amount over to the Indiana return to calculate as refund. The problem is that the state may delay the refund, question it, maybe initially deny it, and then you have to spend extra time and effort to prove everything so that Indiana refunds you.
The other approach would be to not include the information regarding the extra payment but instead contact the Indiana DOR. You are correct that you don't owe the money, but even if it is expeditious, it will still take a little bit of time to get Indiana to process it.
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