Actually, if there is property in Ohio but not in Florida, the probate estate should be set up here. I don't know how much we're talking about but you may be able to do a relief from administration that would be quick and easy to do (you could actually do it yourself) and get the authority that way.
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You know that you are the heirs, but the State doesn't know that. As far as they know the deceased may have had a Will giving everything to the Man in the Moon. You'll never get anywhere with them until you provide the documentation they need and request. Contact a Florida probate attorney in the county where the deceased lived and ask about a summary distribution affidavit or what ever form they use on a 10 year old probate. You can locate a Florida probate attorney using the Find a Lawyer search tool on AVVO.
Both Mr. Davis and Mr. Reynolds have given you sound answers. Your Power of Attorney is irrelevant. You must establish that you are the next of kin. The personnel at Ohio Unclaimed Funds should tell you what they will accept in the absence of a FL probate, but you are going to have to prove one way or another who are the "next of kin" for inheritance purposes. I suspect an affidavit and copies of the birth certificates of each child might do it, or even an obituary from the paper that indicates the next of kin accompanies by driver licenses ... but they have the money so they set the rules. If you are not good at this type of detail you may need to hire an attorney to get the funds.
Mr. Huddleston is an Ohio-Certified Specialist in Estate Planning, Trust & Probate Law, with offices in Columbus and Dayton, serving client families and private business owners throughout Ohio. He may be contacted directly by phone toll-free at 888.488.7878 or by email CLH@HUDDLAW.COM. Mr. Huddleston responds to Avvo questions as a public service to help educate and provide general guidance to questioners, but his responses are not legal advice and do not create an attorney-client relationship.