Here's a more detailed summary of the options to go from Khiva to Nukus, based on our experience a few weeks ago.
There is no marshrutka or shared taxi going from Khiva to Nukus directly, at least not in early November.
Private taxi: the standard quote we got from many different persons was $50 (for the whole car), which we negotiated down to $30. We still didn't use this option as it was only the two of us and therefore too expensive.
From Khiva north gate, there are minibuses (4,000/seat) and shared taxi (7,000/seat) going to Urgench. There is also a trolleybus apparently but it goes way slower for the very little money you may save
In Urgench, you will arrive at 'Dekon' bazaar. From there, you can take a taxi to the other bazaar or the Olympic stadium, where you will find shared taxi going to Nukus for 40,000-50,000/seat. Otherwise, you can take a minibus from Urgench Dekon bazaar (exact location: Конечная городских авттобусов - http://ge0.me/ozs8Dx90j1/Конечная_городских_авттобусов
) to Beruni (3,000/seat), where you can then find a bus (20,000-25,000/seat) or a shared taxi (30,000/seat) to Nukus.
We used this second option (via Beruni) as we thought it may be both easier and cheaper than to take a taxi from one bazaar to the other in Urgench, but in fact the minibus to Beruni is very slow and we had to wait an hour in Beruni to find other passengers to Nukus. If I had to do it again I would take a shared taxi from Urgench to Nukus directly. In Nukus, the shared taxi drop us directly at the Savitsky museum.
In Nukus, I can recommend Besqala hostel. We paid $10/bed in a dorm (breakfast included), which seems to be an ok price in Nukus (the guy would not go lower). It was very clean, quite conveniently located and breakfast was decent. They can arrange a taxi to the border for $12 (negotiated down to $10) per car, which is a convenient option for an early start.