That's strange that the lender doesn't have a copy of the application, a friend of mine had a similar problem and was told the bank didn't have a copy and to go back to the broker etc. long story short is that the bank did have a copy, it was in the archives and a bit time consuming for them to retrieve. I also thought from an audit point of view the bank should at least have a scanned copy of same.
The application will definitely be required as the MIS will need to see the amount of mortgage you signed the application for credit for.
I would suggest going back to the bank, speak to someone senior and ask them to check their archives.
As regards the broker I have no idea what happens to their documentation. Sorry