Oh God, as an Australian who lived in London I really feel for you. It is undoubtably the worst national characteristic. The number of times I had some friend of my brother's or someone I hadn't seen since school lob up on my sofa and whinge about how expensive London is (I KNOW it's expensive! I LIVE HERE! How could it POSSIBLY have escaped my attention? ). I actually had to tell my mum not to give out my number to anyone saying they were going to London and wanted to look me up - "look me up" always meant "stay for six weeks for free".
I think you're going to have to grasp the nettle and tell her to cough up or bugger off. No point being subtle - sit her down with the flat share bit of Loot, a red pen and a phone. I think she owes you, actually. Asking her to babysit on NYE is a big deal, but it's definitely not out of proportion to the favour you're doing her.
An Australian friend of my husband's turned up for weeks when we lived in Hong Kong. He never cooked or contributed financially, and I came over all passive-aggressive and started making pasta and garlic bread every night because he was allergic to gluten. Get your own farking dinner, love.
In fact now we are in Shanghai (another expensive city that everyone wants to visit for free), I think we are having a baby just to fill the guest room. I'm really looking forward to not having a "spare room", only a nursery!
Never, ever buy a sofabed if you know any Australians would be my number one tip for modern urban living.