Deposit and rent in the same bank account?

[Kia Zhu]

My property is in NJ. Per the bank's request, I need to ask tenants to fill in a W-9, so I can put their security deposit in an interest bearing security deposit account.

The tenant wouldn't fill one, so I put her security deposit in the same account my tenant deposit their rent.

This tenant broke the lease, and left 8 months prior to her lease ends (although she gave 30 days notice). I gave her the remainder of her security deposit (I was able to find a tenant after she moved out and only deducted half of April from her security deposit since the new tenant moved in the middle of the month).

Now she is not satisfied with returned deposit and requests proof that her security deposit was not commingled with personal. I am not sure If I did it ok here. How do you guys handle this kind of matter?