If you suspect that he is hiding earnings in the deceased's account I would possibly seek legal advice before you do anything, even informing the bank that she is dead, as if he finds the account closed or frozen he will know to hide the income elsewhere and you may have missed your chance?
retirednow I reported it to the bank a while ago when i found out because I found an old statement dated 2014. Then there were no more letters (the post situation is another complicated issue). Anyway, nothing came through the post to my knowledge and i assumed the bank had sorted it. I was told back then they couldn't give me any updates because of confidentiality. No statements came in the post, maybe he's moved it to online banking. But yesterday a new bank card turned up. I've had a new bank card recently and it was still in it's envelope. Same bank, same envelope.
If you live in the same house, can't you find the death certificate and post a copy of it to the bank? When my dh died, I informed the bank and they froze his account immediately, but I did have to show them the death certificate and the bank account was in his sole name.
The bank don't necessarily need the death certificate. I work in financial services and we use various services to identify where customers have died and we've not been notified. I would take the unopened letter to the bank and write across the envelope that she died 6 years ago. I'd also write a letter, addressed to the financial crime manager of the bank/branch outlining your suspicions and say you will contact the regulator if you don't get a receipt of your letter as a minimum. They won't be able to divulge the details of any investigation they undertake, but they should acknowledge receipt of your correspondence.