Efficient verification of Kemeny-optimal rankings
The following decision problem is NP-complete:
Kemeny Score
Input: An election ($n$ partial or full rankings of $m$ candidates) and a positive integer $k$.
Question:: Is there a ranking with $\sum_i d(r, q_i) \leq k$?

This decision problem is often seen as "natural decision problem" behind Kemeny voting. It is trivially in NP since computing the Kendall-Tau distance is in P. For NP-hardness see for example [Bartholdi, Tovey and Tick, Social Choice and Welfare 1989] or [Dwork, Kumar, Naor, and Sivakumar, WWW 2001].

However, the problem you describe above looks more like an optimization problem. To answer your question I refer to the Kemeny Winner problem which is $P_{||}^{NP}$-complete (see [Hemaspaandra, Spakowski, Vogel, TCS 2005]). Thus, already deciding whether a candidate is an (optimal) winner in Kemeny's system is presumable not in NP.