Hi Reema,
Sorry to hear about you dad.
I submitted my mum's visitor visa application saying we want to be together in our dad's 12th death anniversary. Plus, she was living alone back home. Her visitor visa was approved for six month. So, I think being true and honest with the reasons is good. They will understand it. You also have your brother under 18, so that helps.
If you are resident, you could sponsor your mum and your documents will be essential. But as you are in work visa (the way I am), your mum should be able to show enough funds with her to travel and live. About the $15,000 joint account, it is essential to demonstrate that the funds can be used by your mum. May be the bank can write a cover letter saying that your mum is entitled to withdraw the fund, or she can transfer the fund to her individual account and attach both bank statement with explanation. The other options is to send fund from your account to your mum's bank account and submitting her bank statement and fund transfer evidence.
You can say that there will be no accommodation cost as she will be living with you. For this, you need to attach house contract, your bank statement of around last three months (highlight the salary/wage you get and rental paid) and pay slip, IRD statement will strengthen.
You should definitely attach your cover application.
Hope this helps.
Thanks,
Saumya