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Any accounts that husband was on jointly with mother would now be his solely owned accounts along with anything in them. A joint owner's interest automatically extinguishes upon their death leaving the survivor the sole owner.
As for the other account in her name only, that would now be in her estate and would be subject to going through probate. If her entire estate is under $60,000 it would likely qualify for "family administration" which is an abbreviated probate process that doesn't require all the formalities of a full probate case.
The checks would also be part of her estate and the petitioner in a family administration case could deposit them into her bank account and then request that the entire account proceeds be delivered to him/her.
This is a link to the statutes and the general process for a family administration in FL:
Link: FL Family Administration