Basic testing is done from a urine sample, which can be collected from any pathology clinic. The tests are carried out at the RPA Hospital or ARL in Melbourne.
The test is not covered by the government but only costs some $30-$40. A doctor’s script is required for the RPA test, and any practitioner registered with ARL can order a test.
The report is easy to read, it shows whether you are severe, moderate, or mildly deficient or OK.
Correcting the problem:
This depends on the severity of the problem. If severe it should be managed by a qualified practitioner.
In general, buy and use iodised salt. Kelp has been a good source of Iodine but studies have shown that it also contains bromine, which may negate the benefits.
Iodine solutions are available, but frankly do not taste too good. Incidentally, normal Iodine aqueous solution may be used, but very sparingly.
There are also tablets available, unfortunately not common in Australia - but obtainable.
For those interested in studying this matter in more detail the references for all four articles in the series:
PubMed
16489900 – Are Australian Children Iodine deficient?
Regarding Iodine and estriol
PubMed
17119638, 16453033, 15692619, 15162225, 12828370.
15239792 – the effect of supraphysiological levels of iodine on patients with mastalgia.
8772562 – Relationship between breast cancer and thyroid disease.
Recommended Books:
What your Doctor may not tell you about Breast cancer – John R Lee MD
Iodine – Why you need it, why you can’t live without it – David Brownstein MD | 
