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The deed is not legally effective until the filing date as that is the date that the public is put on notice of the transfer. Thus, legally, the property didn't legally transfer until December 2011 - when the deed was recorded.
However, if your mother collected rent personally - and not through the LLC - she should include the income on her personal income taxes. Also, if the LLC didn't make any money, there's likely no reason at all to file a return. However, I would defer to a CPA for the tax issues - it may be that a CPA would want her to file a return showing no income and no assets.
As for dissolving the LLC, you can fill out the dissolution paperwork and file it with the Florida Secretary of State. Here's a link to the forms you need: http://form.sunbiz.org/pdf/cr2e048.pdf
Sorry for the confusion. The comment that "if the LLC didn't make any money, there's likely no reason at all to file a return" is in regard to the need for the LLC to file a return. Thus, I don't think it would be necessary for the LLC to file a return if it made no money.
She would file a return personally, however.