It took me a long time to find work too in good part because I specialise in something that wasn't in much demand where I live when I first moved over here but also because of the visa challenge. The first job offer I had was reneged as soon as it was offered when the person I spoke to found out that I wasn't a permanent resident at that time (I am now). I can laugh at it now because they were prepared to hire me without an interview but I felt frustrated and a little bit defeated at the time. I signed up with lots of different agencies and found a really, really good one that understood my skill-set and career background and I was soon picking up short and fixed term contract work. One of those contracts turned into an offer of permanent work and I am still there now.
This is what worked for me:
1. As others have said, take your time over your application. If no selection criteria was requested, I wrote one anyway using my best guess at what they might require.
2. If you're open to trying contracting or temp work and plan to try an agency, ring round until you find someone you think you click with.
3. Check your C.V. and write out any acronyms and initialisms in full even if you worked for somewhere that would be obvious to a fellow Brit such as the NHS, HMRC or HBOS. Remove any reference to your residential status to increase the odds of being asked for an interview. Not that you won't be asked to prove that you are eligible to work in Australia but that you don't always have to provide it in the initial letter of application.
4. Talk about what you do to as many interested people as you can. Chances are that someone is bound to know something about the industry you want to get into or at least have some useful working knowledge of the job market in your area.
It took a long time to land the kind of work I love, working for the kind of business and people I like to work for but I did manage to get there in the end and I sincerely hope that you do too.
Best of luck.