There are a couple of other ways you can get to Ilkurlka apart from travelling the full length of the Anne Beadell from Coober Pedy to Laverton.
One way is to travel from Nullarbor Roadhouse on the Eyre Highway via Cook and Vokes Hill to Ilkurlka and then westwards - this is a good option when the Woomera Prohibited Area is closed to traffic. This trip can be combined with a visit to Maralinga where it is possible to tour the nuclear test site (advance bookings required). See here for further details.
Another option is a good one for an excursion into the Great Victoria Desert from Kalgoorlie. The journey takes you along the Transcontinental railway eastwards to Kitchener, north to Neale Junction, east to Ilkurlka and back via Laverton to Kalgoorlie. A fairly easy drive into the desert which could be done in three or four days. See here for further details.
A nice and easy option is a round trip from Laverton to Ilkurlka - plenty to see and experience and it could be done in two days. See here for further details.