We did a combination of what PP's are suggesting - we spent 16 days in Hawaii, with time in Maui, Kauai, Waikiki and 6 nights at Aulani. The Aulani portion of our trip was last and I planned sightseeing for every other day so we would have ample time to enjoy the resort. We wanted a combination of exploring Hawaii, relaxing, and enjoying Aulani which was a destination for us, and I think we accomplished that.
I do agree that you should rent a car and spend a lot of time exploring Oahu, whether you do a split stay on the island or stay at Aulani the whole time. We love Aulani but I think one should have a desire for a Hawaiian vacation rather than an Aulani vacation, if that makes sense. We didn't think the beach or resort alone was enough to offset the cost and travel involved, if you aren't also interested in the rest of Oahu or Hawaii. Even Disney constructed the resort under the assumption that Aualni would be a place to stay for Hawaii vacationers, as opposed to Aulani being the main destination.
As for budget, that's hard to say. We are two adults and flew from the east coast in coach. We spent 3 nights at a Marriott in Waikiki, 3 nights at a Westin in Maui, 3 nights at a Marriott in Kauai, and 6 nights at Aulani. We did a lot of excursions and didn't cook our own meals. Our total, including money spent there, was about $15,000. If we go again, it would be a shorter trip with fewer excursions so I'd budget about $7000 for it. We'd likely do 10-12 days next time and we did find that it took about 2 full days to adjust to the time difference, where we weren't zonked by 8pm and up at 4am.