<p>Say I have a linked list of numbers of length <code>N</code>. <code>N</code> is very large and I don’t know in advance the exact value of <code>N</code>. </p>

<p>How can I most efficiently write a function that will return <code>k</code> completely <em>random numbers</em> from the list?</p>

I would suggest: First find your k random numbers. Sort them. Then traverse both the linked list and your random numbers once.
If you somehow don't know the length of your linked list (how?), then you could grab the first k into an array, then for node r, generate a random number in [0, r), and if that is less than k, replace the rth item of the array. (Not entirely convinced that doesn't bias...)
Other than that: "If I were you, I wouldn't be starting from here." Are you sure linked list is right for your problem? Is there not a better data structure, such as a good old flat array list.