@DebtAfterDeath As others have said, any debts are paid out of the late person's estate, before anything else is paid to any 'beneficiaries' and your SIL should NEVER have just divided it up between her and your husband until all income and debts were established and debts settled, or at least as far as possible.
The advice about putting a notice in the paper is probably a good one if you can't be sure what debts he has - but you may find that there is sufficient paperwork in his house to let you know without doing that. The expense of doing that will need to come off the £6000 and not your personal money. Your SIL and DH need to 'reunite' the money received - and add it to the £10k if you get it from the Life Insurance Policy. Also advise Royal Mail to redeliver any mail addressed to him for, say, 6 months
Personally, although I used the 'tell us once' option, I also did phone all the people who needed to know as I was anxious to deal with things as speedily as possible and I didn't want to wait, possibly for weeks, to hear back from various sources
You may find ( as I did, to my surprise) that when you tell people regarding his regular commitments - Council Tax, Energy company, Any Broadband or telephone companies etc etc that there may be refunds/rebates come into the Estate - or you may find that Money is owed ( like outstanding pension payments). You won't know what needs to be paid out until that is done and you should wait and see until you have heard from all these people before going forward.
The bank didn't do anything wrong in paying the money out as it was only a small amount in their eyes ( individual banks have different ideas about when they want Probate) - but your SIL and DH need to make sure that they follow the correct procedure now or they MIGHT find themselves liable for payment of outstanding debts, months and months down the line. I am in Scotland and the advice here was to wait 6 months after My Mother died before distributing anything that was left, just to ensure nothing unknown turned up
Either SIL or DH need to keep written accounts of what comes in and goes out. Doesn't need to be anything fancy though, just the details - and also keep any receipts relating to any expenditure. You may have needed to put more towards his funeral too, depending on what his funeral plan covered and that would come off the £6,000 if it hasn't already