Rishi, you said you'd lost your job recently. I'd suggest that trying to tackle the home situation right now might be too much for you to take on alone. You've lived effectively separate lives, under the same roof, for some time - a couple months more won't make that much difference. If this lady's aim is for you to bank-roll her lifestyle, then if you have to make savings she might clear off anyway, or she might change (doubt that myself).
Why not use this time to get some talking help for yourself - invite the missus to do the same, but don't hold out expectations. I think you can get referrals to therapists on bulk billing in Australia. Just being able to talk to someone neutral should, like coming on AB, be a source of comfort at least, if not of practical advice.
From what you've described, trying to confront her on your own isn't going to work.
If she's been neglecting the kids as long as you describe, they're going to be fairly used to doing stuff - so rather than taking on tasks like 'putting the kids to bed', enrolling the kids into a big joint adventure of 'let's sort X out, let's make Y for dinner' might sweeten things.
Best of luck old dear.